Issue 
Manufacturing Rev.
Volume 5, 2018



Article Number  2  
Number of page(s)  26  
DOI  https://doi.org/10.1051/mfreview/2017014  
Published online  16 March 2018 
Review
Modelling of additive manufacturing processes: a review and classification
Laboratory for Manufacturing Systems and Automation, Department of Mechanical Engineering and Aeronautics, University of Patras,
Patras
26500, Greece
^{*} email: pstavr@lms.mech.upatras.gr
Received:
6
October
2017
Accepted:
21
November
2017
Additive manufacturing (AM) is a very promising technology; however, there are a number of open issues related to the different AM processes. The literature on modelling the existing AM processes is reviewed and classified. A categorization of the different AM processes in process groups, according to the process mechanism, has been conducted and the most important issues are stated. Suggestions are made as to which approach is more appropriate according to the key performance indicator desired to be modelled and a discussion is included as to the way that future modelling work can better contribute to improving today's AM process understanding.
Key words: Additive manufacturing / modelling / state of the art / process parameters / key performance indicators
© P. Stavropoulos and P. Foteinopoulos, Published by EDP Sciences 2018
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Additive manufacturing (AM) namely the process of joining materials for the production of objects, made of 3D model data, usually layer upon layer, as opposed to subtractive manufacturing methodologies [1] are applicable to a wide range of materials, including metals, composites [2] and even biomedical products [3]. AM [4] differs from rapid prototyping in the fact that AM specifically aims at the manufacturing of end user parts, rather than just prototypes [5]. The interest in AM processes has been steadily increasing in the last years and according to estimations it could exceed 5% of the total global market [6]. AM technologies have issues (Fig. 1) related to low productivity, poor quality and uncertainty of the final part's mechanical properties [7].
In this paper the AM modelling is classified according to key performance indicator (KPI), process parameters and the modelling approach/analytical, numerical or empirical [8]. For this study a classification [9] according to the process mechanism (ISO 172962) [10] has been followed (Table 2).
Fig. 1
Issues of AM. 
Nomenclature.
Classification of AM processes.
2 Classification of modelling for AM processes
In this section, the modelling approaches of each process group are classified according to the KPIs and the process parameters used. It has to be noted that the modelling approach followed by some studies does not aim to the connection of the process parameters to the simulated KPIs. In such cases, the table cell referring to the process parameters is left empty, highlighting the KPIcentered perspective of those approaches. The most important issues of each AM process group are also described.
2.1 Vat Photopolymerization processes
The parts created through Vat Photopolymerization (VP), tend to have high dimensional accuracy and surface finish when compared with the majority of other AM processes. Moreover, the building time is an advantage of the VP technologies that use mask projection, in which an entire part cross section can be projected. The main drawback of the VP processes is their use of photopolymers, their impact strength and durability, which are inferior to those of good quality injection molded thermoplastics [11]. As a result, the main issue of the VP processes is that of the manufactured parts' mechanical properties and even though this is more an issue of the material type used, the optimization of the process, in terms of mechanical properties, can lead to improvements in this field. In Table 3 the process parameters and the KPIs of each modelling approach can be seen.
More specifically, the modelling works of VP focus on the topology and dimensional accuracy ([12–14] numerical) and mainly on the mechanical properties ([16] analyticalempirical, [17–21] numerical and [21–24] empirical) and finally in [25] an analyticalempirical approach models heat transfer related issues.
Classification of the of the modelling studies on the VP AM process.
2.2 Powder bed fusion
In the powder bed fusion (PBF) process group, there are four different fusion mechanisms: the powder particles are fused together with the use of solidstate sintering, chemically induced sintering, liquidphase sintering or full melting [26,27]. In this study, the division among the PBF processes will be made in three subcategories in order to address not only the difference in the process mechanism (sintering and full meting), used by the most commercially available machines, but also the difference in the energy source (laser or electron beam). As a result, the three PBF subcategories that will be used in the categorization of the existing studies, are the selective laser sintering (SLS), selective laser melting (SLM) and electron beam melting (EBM) process groups.
The high residual stresses that are present in the PBF AM processes, especially for metal manufacturing, lead to the warping of parts. In order for that to be minimized, techniques, such as the use of internal cooling channels, the careful selection of the part's orientation and the location of the supports have to take place. However, in order for that to be possible, the effects of the part's thermal history (residual stresses and thermal distortions) have to be taken into account. As a result, the modelling of thermal and thermomechanical phenomena, taking place in the PBF processes, are of crucial importance for the optimization of the processes. More specifically, the combination of the laser power, spot size and scan speed determines the fusion depth whilst the melt pool dimensions have a direct impact on the residual stresses of the parts. Powder shape, size and distribution strongly influence the laser absorption characteristics, as well as the powder bed density and powder bed thermal conductivity. As a result, those parameters have to be taken into consideration. Moreover, the selection of the laserpower and the bedtemperature play a crucial role in the dimensional accuracy, density, shrinkage and curling of the produced part, as well as in the recyclability of the unused powder. Consequently, all the above parameters have to be considered in the thermal modelling of those processes [11]. The different subgroups of the PBF family are described below.
2.2.1 Selective laser sintering
There are a lot of studies on the modelling of the SLS process, with an almost equal distribution among the various KPIs. In Table 4 the modelled KPIs and the process parameters that have been used in each study can be seen. Modelling of the topology/dimensional accuracy takes places in [28] (analyticalnumerical approach), in [29–34] (numerical) and in [35] (empirical). Mechanical properties and microstructure modelling has been carried out in [30,36–41] using numerical approaches, whereas in [42] an empirical one has been followed. According to [43], the microstructure, physical and mechanical properties of the parts' being manufactured with the use of SLS, are fundamentally affected by laser power, laser scan speed and laser scan spacing. All of those parameters are directly related to the amount of energy imparted on the powder surface of the printed part. Build time has been analytically modelled in [29] and empirically in [44,45]. Energy consumption has been numerically modelled in [33,47]. Finally, most of the studies model heat transfer related KPIs either simultaneously with other KPIs [28,30,32,40,41,46] or exclusively [48–54] and in either case using the numerical approach.
Classification of the of the modelling studies on the SLS AM process.
2.2.2 Selective laser melting
SLM is a process similar to that of SLS; the two are instantiations of the same concept, but differ in technical details. In the SLM process for a part's formation, powder melting occurs instead of sintering. The modelling works, available in SLM, have a similar distribution to that of the SLS; in Table 5, summarization of the process, parameters and KPIs of the different modelling approaches can be seen.
More specifically, the surface roughness of parts has been modelled in [55,56] by an analytical and a numerical approach respectively. In [31,32,57–61] topology and dimensional accuracy issues have been modelled using the numerical approach exclusively. In [62] mechanical properties and microstructure have been modelled analytically, whereas in [56,63] numerically. Finally, most of the studies model heat transfer related KPIs either simultaneously with other KPIs [31,32,41,48,50,57–59] or exclusively [64–70] and in either case using the numerical approach.
Classification of the of the modelling studies on the SLM AM process.
2.2.3 Electron beam melting
EBM or EBAM (Electron beam additive manufacturing) has various advantages (high energy efficiency, high scan speed, moderate operation cost); however, process stability, part defects and quality variations are some issues that need to be improved [71]. In Table 6, a summarization of the process, parameters and KPIs of the different modelling approaches can be seen.
In EBM, there is a limited number of the existing modelling publications, which focus almost entirely on thermal modelling [72], in which the analytical approach has been followed and [32,41,49,50,73–77] via numerical methods. However, in [32] the residual stresses and distortions are also modelled using numerical methods.
Classification of the of the modelling studies on the EBM AM process.
2.3 Directed energy deposition
In the directed energy deposition processes (DED), the material deposition and melting are performed simultaneously. More specifically, energy is guided to a narrow focused region, where the substrate is melted, when heated by the power source (mainly laser beam), while at the same time, there is deposition of material which, as a result, is also melted [11]. The material can either be in wire or powder form. Most of the DED machines are very flexible concerning the process parameters' selection whilst the effects on many of them are strongly interrelated (powder feed rate, beam power, and traverse speed). In addition, their impact on the melt pool characteristics and on the thermal history determines the warping, residual stresses and the surface roughness of the parts [11]. Moreover, droplet kinematics, like in the MJ process group, play a major role here as well. The above references clarify the importance of modelling of the thermal history of a part, in which the laser power, scanning speed and melt pool characteristics are taken into account. Such studies can be utilized for the selection of the optimum process parameters, as well as for the optimization of the process itself, minimizing the need of expensive and time consuming experimental trial and error methodologies. The thermomechanical effects and especially the fluid dynamics have to be taken into account as well. In Table 7, a summarization of the process, parameters and KPIs of the different modelling approaches can be seen.
There is a plethora of modelling papers on the DMD process which are distributed, almost evenly, among the various KPIs. More specifically, modelling of surface roughness is presented in [78] (analytically) and in [79] (numerically). Modelling of topology and dimensional accuracy takes place in [80–84] using analytical, in [85,86] analyticalnumerical, in [17–19,32,79,87–94] numerical, whereas in [95] those issues have been empirically modelled utilizing artificial neural networks (ANN). Modelling of the mechanical properties and microstructure has been conducted in [80,81] using analytical, in [86,96] analyticalnumerical and in [94,97–101] using numerical approaches. Droplet kinematics and flow phenomena have been modelled in [102,103] using analytical and in [87,98,104–113] using numerical approaches. Finally, heat transfer related KPIs have been modelled either simultaneously with other KPIs (in [114] analytically, in [85,86] analyticallynumerically and in [32,41,49,50,87,90–92,94,98,106,107,112] numerically) or exclusively ([115–118] numerically).
Classification of the of the modelling studies on the DED AM process.
2.4 Binder jetting
Parts created by means of binder jetting (BJ), with plaster based powder and water based binder processes, tend to have low strength and stiffness. This is solved with the use of infiltrants, which greatly improve the material properties. Another strategy is the use of a polymethyl methacrylate powder and a liquid binder that causes a curing reaction, at room temperature. In this case however, after the printing has been completed, the part has to remain in the build chamber for several hours in order for the curing to be completed. In order for metal parts to be manufactured, via the BJ process, a series of post processes is required. More specifically, three furnace cycles are necessary after the printing of a metal part, in order for the binder polymer to evaporate and the part's density to increase. This is achieved with the addition of extra metal ingots, being in contact with the part. Generally, the dimensional accuracy and the surface roughness of parts, made by BJ, are not as good as those made using MJ and they tend to have poorer accuracies and surface finishes [11]. These problems are mostly attributed to the inherent characteristics of the process; however, modelling and optimizing the process mechanics can lead to the improvement of such issues. In Table 8, the modelled KPIs and the process parameters used in each study can be seen. In [119], empirical models of shrinkage rate, surface roughness are presented, whereas [120] and [121] deal with topology issues analytically and numerically respectively. In [122] an analytical energy consumption model has been developed, while a semianalytical approach is presented in [123]. Studies using a numerical approach [113,121] model droplet kinematics and flow phenomena.
Classification of the of the modelling studies on the BJ AM process.
2.5 Material extrusion
The commercial name of material extrusion is fused deposition modelling (FDM). Some of the major issues that deteriorate the quality of parts are analyzed. The cooling process profile determines the part distortions and as a result, material warping can be the effect of nonlinear cooling. Furthermore, the creation of porous parts is possible and it is also determined by the cooling profile. More specifically, the temperature differences among the building platform, chamber and the different layers of the part, play a major role. In addition, the temperature of the extruder and that of the layer on which the filament is placed, determines the success of the bonding between them and as a result, the mechanical properties of the final part [11]. Another important issue is the fact that the creation of the parts' material properties via material extrusion is anisotropic. This is due to the crisscrossing manner used by the material extruder in order to deposit the filament [137]. Thermal modelling issues concerning the material properties and dimensional accuracy, as well as the improvement of other KPIs, namely, building speed and surface roughness, are the main areas that modelling studies on ME concentrate. In Table 9, a summarization of the process parameters and KPIs of the different modelling approaches can be seen. More specifically, surface roughness modelling is more extensive in ME, with the analytical approach of [124], the numerical of [125] and the empirical ones of [126–128]. Topology and dimensional accuracy issues have been modelled in [129,132–134] using analytical methods, whereas in [125,135,136] numerical ones have been used and in [137–141] the empirical approach has been followed. Also, the dimensional deviations, caused by changes made in layer thickness and deposition angle, are analytically modelled in [129].
Moreover, in [130], a FEA model is used for the evaluation of a part's distortions, using a parametric study, for the evaluation of the deposition parameters effects on residual stresses and part distortions. FEA has been used in [135] for the simulation of the mechanical, thermal phenomena, the toolpath effects, the residual stresses and the part's distortions. Also, in [136], 2D and 3D numerical analysis of melt flow behaviour of a representative ABSiron composite, through the liquefier head, has been carried out, whereas in [141], ANOVA has been employed to investigate into the impact of various process parameters on elastic performance. Modelling of build time takes place in [142] in which both an analytical and empirical, whereas, in [125] a real coded genetic algorithm is used in order to obtain the optimum solution, concerning the part's deposition orientation, the simultaneous enhancement of its surface finish and the reduction in build time. Finally, in the studies of [147] and [155] heat related KPIs are also modelled utilizing numerical approaches.
Classification of the of the modelling studies on the ME AM process.
2.6 Material jetting
The technical problems of the MJ process are various. Some of the most important factors are the droplet velocity and size, which play a major role in the deposition characteristics. In addition, the satellite droplets, that break off from the main droplet, during the flight and result in not welldefined boundaries [104], along with the droplet splashing on impact, leading to the formation of a “crown” [105], have to be tackled with, in order for the quality of parts created by MJ to be increased. Consequently, fluid dynamics, in which temperature is also taken into account, has to be used so as to address the most important issues faced by the MJ process group. In Table 10, a summarization of the process parameters and KPIs of the different modelling approaches can be seen.
Most of the studies that refer to the MJ process group focus on the droplet kinematics and the flow phenomena. Namely, [104–106,111,121,143,146–153] in which numerical approaches have been used, whereas in [154] empirical and in [145] analytical ones have been used. In [17–19,121,143] issues of topology have been addressed by means of the numerical methods and in [144] the effects of various process parameters on the mechanical properties of parts have been empirically simulated. Finally, heat transfer related issues have been modelled in [147] and [155] using numerical approaches.
Classification of the of the modelling studies on the MJ AM process.
2.7 Sheet lamination
The different methods used for the bonding of the new sheet on top of the other ones are (a) gluing or adhesive bonding, (b) thermal bonding, (c) clamping, and (d) ultrasonic AM. The adhesive bonding and ultrasonic AM will be presented in the following sections. However, the thermal bonding (sheet metal lamination process) and clamping sheet lamination will not be presented in this study, given that the first has gained little commercial attention [11] and the modelling of the second is beyond the scope of this study.
2.7.1 Gluing or adhesive bonding
The commercial name of this process group is laminated object manufacturing (LOM). The main problems faced by LOM, to some extent, are similar to those of ultrasonic consolidation (UC). In Table 11, the aforementioned studies, along with their process parameters have been summarized. More specifically, if the laser power induces more thermal energy than it is required, the efficiency of the bonding between layers is reduced. Moreover, phenomena of part distortions, due to nonuniform heating and cooling and also edge roughness are common [166]. In order to cope with such problems, most of the modelling studies on LOM are thermal simulations ([164] analytical and [160,165] numerical). A surface roughness model has been developed in [156] through an analytical approach, whereas in [157] and [158] an empirical approach has been followed. In [159], an analytical microstructure model has been developed. Numerical modelling of the mechanical properties has been conducted in [160] and an empirical one in [161]. Finally, two build time models, one analytical [162] and one numerical [163] have been included.
Classification of the of the modelling studies on the LOM AM process.
2.7.2 Ultrasonic AM
In ultrasonic additive manufacturing or UC, which is the most common commercial name of this process, the most common defects are voids created during the fusion of the different layers of sheets between them, leading to the deterioration of the mechanical properties. Those voids can be classified into three different categories: those that are created because (i) of the surface roughness of two consecutive layers, (ii) of damages due to excessive energy input, (iii) of defects between adjacent layers [167]. Moreover, the mechanical properties tend to be anisotropic because of the difference in the mechanical properties between the interior of the metal foils and the areas where bonding between the different foils takes place. This anisotropy is greater in the z than it is in the x, y directions. Finally, the local microstructure also plays a major role in the final part's mechanical properties, considering that some parts of the foils undergo plastic deformation during the process. As a result, the modelling works on UC aim to determine the process parameters that will ensure optimized mechanical properties and a microstructure, with minimization of the void defects and maximization of the linear welding density (the percentage of interface which is bonded, divided by the total length of the interface between two ultrasonically consolidated foils). In Table 12, there is a summary of the aforementioned studies.
More specifically, modelling of the mechanical properties has been carried out in [168,169] using analytical approaches, whereas in [170,171–175] numerical methods have been utilized and in [167,176] empirical ones.
Classification of the of the modelling studies on the UC AM process.
3 Indicative studies for each KPI group
In this section, a KPIcentered perspective is followed. More specifically, three or more indicative modelling approaches, namely analytical, numerical and empirical will be presented for each KPI group, in a more detailed way.
3.1 Mechanical properties and microstructure
In [48], Kovaleva et al. have analytically modelled the internal structure of a loose powder layer of the SLM process. Using the vector equations of motion, they have created a system of five equations, whose solution has provided the positions of the powder particles. On the left of Figure 2, the free body diagrams can be seen and on the right, a sample solution, which has resulted from an analytical solution, is depicted.
Kumar et al., in [98], have created a numerical model that takes into account both the fluid flow and the heat transfer phenomena happening in DMD. The spead of the droplet, the solidification speed and the interlamellar spacing have been simulated. The solidification front velocities, which were calculated by the model, have been imported in the JacksonHunt relationship. In [167], Ram et al. adopt a design of experiments (DOE) approach in order to evaluate the effects of various process parameters on the microstructure and the laser weld density of the UC process. Some of the experimental results used for the creation of the empirical model, can be seen in Figure 3.
A more macroscopic approach to the modelling of a part's material properties, created with the use of ME is presented in [137]. More specifically, the DOE method has been followed in order for the importance of various process parameters of FDM to be determined. A macroscopic approach has been utilized and the parameters examined were raster orientation, air gap, bead width, color, and model temperature. The impact on the tensile and compressive strengths of the aforementioned parameters has been evaluated. It was found that the tensile strength was mostly affected by raster orientation and air gap. In Figure 4, the tensile strength of specimens with different raster angles, with air gap, can be seen and are also compared with the tensile strength of injection molded parts.
All three modelling approaches are suitable for the modelling of this KPI group, however, analytical models tend to be complex and capable of dealing with very specific cases. However, if the set of assumptions are carefully selected keeping in mind the exact aim for which the simulation will be used, they are capable of fairly accurate and fast to run simulations. Such analyses can be used as a first approach for a problem, or even for process control, for which the combination with experiments and empirical equations can contribute to an increase of the accuracy of the methods. The numerical approach is capable of producing very accurate results for this KPI group, however, they require a longer developing and optimization and require long run times.
Fig. 2
Left: Free body diagrams of different types of motions of the powder particles [48], Right: Results of the model of [48]. 
Fig. 4
Tensile strength specimens with various raster (−0.003 inch air gap) compared with injection moulded parts [137]. 
3.2 Dimensional accuracy
Thermal models have been created in order to model the part's cooling, which is responsible for the residual stresses and thermal distortions. Thermomechanical models have also been created in order to enable the prediction of thermal stresses and distortions, which may lead to the loss of quality due to the deterioration of the dimensional accuracy or even to a total component failure (unacceptable distortions or breakage). Indicative models using an analytical, numerical and experimental approach follow. An analytical approach was made by Lalas et al. [82] and Salonitis et al. [83] for the calculation of the geometrical characteristics of the produced part. More specifically, the surface tension theory has been applied and geometric relations have been used for the calculation of the following equations, which estimate the geometry of the parts, manufactured via DMD: (1) (2) (3)
In Figures 5 and 6 the symbolisms of the equations can be seen.
In [28], the finite difference and finite element approaches, in combination with analytical expressions of the thermal properties, have been used in order to model the temperature history in the SLS AM process. Its impact on the part's shape has also been taken into account. The boundary condition used is the prevention of heat loss from the free surface boundary, while all the others were held at ambient temperature. The thermal conductivity and heat capacity were assumed to exhibit a linear variation in temperature. The variation in density in the z axis has also been taken into account and calculated via a viscous sintering law, presented in the paper. The variation in thermal conductivity with density follows an experimentally created equation. This model has been utilized for the calculation of the change in dimensions, due to the thermal phenomena that take place during the manufacturing of the part.
In [131], the Taguchi method for the design of experiments was used for the study of the effect on the dimensional accuracy of the ME process of layer thickness parameters, part build orientation, raster angle, raster to raster gap (air gap) and raster width, having taken into account the build orientation.
The fact that this KPI group is directly connected with thermal phenomena, for most of the processes, render the numerical approach as the most suitable one. However, analytical simulations can be very useful for certain cases, like a fast first indication, and the empirical approach can be utilized for accurate results, but for specific cases.
Fig. 6
Clad formation. a) First stage: liquid clad, solid substrate. b) Final stage: liquid clad and substrate. 
3.3 Surface roughness
In the analytical model of Gharbi et al. [78] for DMD, each meltpool is considered as the sum of two semiellipses (one for the upper part, and one for the lower part) on a 2D cross section. This model predicts the roughness, based on the analytical solution of the equation that follows. Details about the symbolisms can be found in Figure 7. (4)
In [56], Chunlei et al. have developed a 3D CFD model for the SLM process, to simulate the interaction between the laser beam and the powder layer, which takes into account the splashing of molten material and evaporation. As a result, features of dents/discontinuities have been calculated on the top surfaces due to the melt recirculation and splashing. This model is enabled to predict the roughness of a part's surface. In Figure 8, the results of a sample analysis using the developed model can be seen.
In [127], Anitha et al. have used the Taguchi techniques in order to analyze the effect of different process parameters on the surface roughness of components, produced by the ME process. Their goal is the minimization of surface roughness, whilst the process parameters used are: layer thickness, road width and speed of deposition. It was found that the most important factor was that of layer thickness.
In the processes that a melt pool is created (PBF, DED) and even more so in the powder jetting applications of DED, a CFD analysis, coupled with a thermal model is required in order to obtain accurate results of this KPI, since the phenomena that take place are coupled and are dynamic in space and time. However, with careful assumptions, analytical solutions are also possible. Finally, the direct way that roughness can be measured is in favor of empirical models, which utilize experimentation.
3.4 Building speed
In [29], the build time of the SLS AM process has been analytically modelled. The full time that is necessary for the part's creation is calculated by summing up the time required for each layer. The time required for the creation of one layer has been divided into the scan and setup time (the second one can be obtained from the machine's manual). Using the following equation, the time it takes for the scanning of a layer is obtained by: (5) where L_{d} is laser scan distance, L_{v} is the laser scan velocity and T_{l} is the scan time of a layer. The total scan distance within a layer, can be obtained from the hatch file. The velocity has been estimated by: (6)where P_{l} is the laser power, R is the reflectivity of the mirror, ρ is the material density, d_{b} is the laser beam diameter, l_{m} is the machine layer thickness, C_{p} is the specific heat, T_{m} is the melting temperature, T_{b} is the bed temperature, k is the sinter factor and L_{h} is the latent heat. The setup time refers to the required time for the laying of each new layer and every other machine function when it is not sintering: (7)where t_{wd} is the time required for the workbed to move down, t_{d} is the material deposition time, t_{wr} is the time required for the workbed to rise up and t_{h} is the time required for the material to be heated. Finally, the build time can be calculated by: (8)where, N_{l} is the total number of layers.
In [163], an octreebased algorithm has been presented for the determination of the laser's path, for the LOM AM process's crosshatching operation to be performed. The results of this numerical calculation have been utilized for the estimation of the build time of the process. More specifically, the total laser path, with the use of the variablesize cross hatching presented before is: (9) while in the case of a fixed crosshatching, equation (11) is used: (10)where m is the total number of layers, the sum of the length of all the line segments of the i layer in the L_{X} list is len(L_{Xi}), the V and H suffixes refer to vertical and horizontal lines respectively, in reference to the laser path, generated for the variablesize crosshatching, A refers to the lines in the actual profile, while F to the filler lines. Finally, FCV and FCH are used in order to indicate the vertical and horizontal lines respectively, for the fixedsize cubes. The improvement in the build time can be estimated by the reduction in the path length when the proposed algorithm is used for the determination of the laser path. The use of the proposed numerical solution for the minimization of the tool path, leads to its reduction in the build speed, by reducing the tool path, which is analogous to the build speed. This reduction is analytically calculated, and as a result, this modelling approach is a hybrid one.
In [44], an empirical approach, via ANN, has been utilized for the creation of a model, capable of estimating the build time of the SLS AM process. More specifically, a multilayer perceptron architecture, using the Levenberg − Marquardt algorithm, has been utilized. The most useful input parameters were identified through a series of correlation analyses and the parameters: zheight, part volume, and boundingbox volume have been selected.
The learning and generalization capabilities of an ANN, and not those of specific programming, mainly determine the accuracy of the results. Consequently, the training and learning algorithms that will be used are of crucial importance. Here, the following degree of similarity is used in order to determine the cases included in the training sample of the NN: (11) (12) where a_{n} is the value of attribute n in case a, b_{n} is the value of attribute n in case b, sim is the degree of similarity between the two cases a and b, regarding attribute n and W_{n} is a weighting factor for the feature n. In the following picture, the errors of the model, presented in [44], are compared with those of [45] and [177], in which empirical and analytical build time modelling approaches are also presented (Fig. 9).
The analytical approach is the most appropriate for the modelling of this KPI, followed by empirical modelling.
3.5 Energy consumption
Even though the aforementioned issues are of greater importance for the time being, the optimization of energy consumption of the AM processes is closely related to cost and as a result, it should also be analyzed. In [47], an analytical model, presented in [46], is used for the calculation of the energy consumption in the SLS process. According to [46], the laser energy input, required to sinter the powder of one layer is: (13) where e is the energy per unit area, a_{R} is the absorptivity of the powder, I_{avg} is the average intensity of the laser beam, B is the laser beam radius as measured on the powder bed and V_{s} is the scanning speed of the laser. The total energy can be calculated by summing the energy required for the creation of each layer: (14)where n_{s} is the number of slices and A_{i} is the surface area of the ith slice. In this equation, the laser intensity is substituted in order to be related to the laser power and the sum of the surfaces of the layers, while is termed as the total area of sintering (TAS). A numerical method for the calculation of a part's TAS is then proposed. As a result, the approach of [46] to the calculation of the SLS AM process's energy consumption is a hybrid analyticalnumerical one.
In [123], an analytical equation for the energy consumption has been used: (15) where i is the number of subprocesses, t_{i.S} is the start and t_{i.E} is the end time of each subprocess and P_{i}(t) is the electrical power required for the subprocess i. More specifically, the different subprocesses defined are: (i) the drying of a printed layer, (ii) the printing of a new layer and (iii) the spreading of a new layer. In the following equation, t_{p}, which is the operation time of printing is calculated: (16)where the Y variables indicate the movement of the printer head along the yaxis, during the binder deposition, n is the number of repetitions and t_{0} is a preparation time. Utilizing experimental results and using a linear regression, the previous analytical expression is modified and the following is calculated: (17)
The time required for the printing of a new layer is analytically calculated, by having utilized experimental data for the results' calibration and higher accuracy. The time needed for the spreading of a new layer is calculated by: (18) where V_{s} is the spread speed, V_{rs} is the rapid transverse speed and X_{left}, X_{right}, X_{lmax}, X_{min} are geometry parameters of the layer. The power required for each subprocess is experimentally measured and a mean value is calculated. As a result, the approach of [123], for the calculation of the SLS AM process's energy consumption is a hybrid analyticalempirical one.
For this KPI, which is directly connected to the build time, the analytical approach is, together with the empirical one, are ideal for its simulation.
3.6 Droplet shape
In [102], an analytical approach is used for the modelling on the shape of droplets that collide with the substrate and the effect of speed of the droplets, of the temperature and roughness of the substrate is taken into account. More specifically, assuming a loss of kinetic energy due to the freezing of the droplet on impact, starting from the energy balance equation the authors substitute the analytical expressions of the kinetic and surface energy of the droplet, measuring the contact angle from the experimental photographs and using an analytical expression for viscosity, the following expression for the maximum spread factor is obtained: (19) where We is the Weber number, Re is the Reynolds number and s* is the dimensionless solidified thickness. In Figure 10 the dimensionless solid layer thickness as a function of contact resistance is depicted, whereas Figure 11 the thermal contact resistance as a function of surface roughness, as calculated by the model of [102], can be seen.
Another analytical work is that of [103], in which transition temperature at which splashing disappears is calculated: (20) where Ste_{c} is critical expression for the Stefan number which is also calculated in [103], H_{f,d} is the latent heat of fusion and c_{d} is the specific heat. In Figure 12 the graph of transition temperature versus impact velocity can be seen.
A numerical approach on the droplet kinematics is presented in [104]. More specifically, the equations of the conservation of mass and momentum, which govern the flow within the liquid phase following impact have been discretized in a typical control volume formulation and solved numerically, using as boundary conditions the surface tensioninduced pressure jump at the droplet surface, a zero tangential stress condition at the surface and the dynamic contact angle at the contact line of the solid, liquid, and gas phases. The explicit solution scheme has been used for convective, viscous, and surface tension effects, whereas pressure has been implicitly solved. Also, the deformation of the free surface is tracked using a piecewise linear volume tracking algorithm. An important characteristic of this study is the use of the continuum surface force model for the surface tension. Finally, the approach for the initiation of perturbation that has been followed is that the disturbance is not imposed on the moment of impact, but later, when the diameter of the droplet is just less than its initial diameter and a sufficient length has been acquired, so as to adequately resolve the perturbation that is imposed. This approach is practical and yields realistic results as can be seen in Figure 13.
In [154], the effects of driving time, volume in the pressure chamber, nozzle plate thickness and volume factor (process parameters) on the droplet volume and droplet velocity (KPIs) have been empirically modelled. In this study, the droplet size characteristics before their impact are modelled, which are determined by a dropondemand piezoelectric droplet generator.
From the above studies it can be concluded that for this KPI group all of the different approaches (analytical, numerical, empirical) should be utilized in the applications that each one is more suitable, so as to produce different results and to be capable to utilize all three complementary to each another.
Fig. 12
Transition temperature as a function of impact velocity. Aluminum substrate with impact velocity of V_{0}, R_{c} contact resistance at the droplet–substrate interface. 
Fig. 13
Photographs and simulation results of the numerical model of the study [104]. Numbers at the right indicate milliseconds following impact. 
3.7 Heat transfer related KPIs/phenomena
An indicative analytical approach is that of [114], in which the calculation of laser attenuation and powder temperatures at every point below a nozzle of a DED system (Fig. 14) is presented . The beam intensity is calculated at any position and it is utilized for the calculation of the temperature of the powder (Fig.15). The space below the nozzle is divided in two: that which is above the powder attenuation plane and that which is beyond; a different analysis is performed for each case. In the first part of the space, above the attenuation plane, the powder particles are divided between those that fall within the laser beam and those which fall outside. In the space beyond the attenuation plane, the powder particles are divided in three categories: those that are continuously under the laser beam, those that were in the beam at the consolidation plane but are beyond when in their final position and in those that undertook no further heating after the consolidation plane. For those five categories a different analytical equation for their temperature is provided.
In [118], a thermal analysis is presented, as part of the simulation of the Weldbased AM process. More specifically, the heat transfer equation is solved in workpiece taking into account both arc and induction heat input: (21) where ρC_{p} is the volumetric specific heat, λ is the thermal conductivity, Q_{arc} is the arc heat term, and Q_{jh} is the average induction heat rate over an electromagnetic period. A combined radiationconvection heat transfer coefficient is used as a boundary condition for the surfaces of the workpiece. A FE discretization is utilized, the latent heat of phase transformation is taken into account by increasing the specific heat and the material properties are temperature dependent. Also, the increased convection of the melt pool due to fluid flow is simulated, by increasing the thermal conductivities by 10 times. Also, through a coupled electromagnetic analysis, which takes place only at a specific time so as to decrease computational costs and times, the induction heating is also taken into account in the thermal analysis, as preheating or postheating. Consequently, the concept of this approach is the creation of a simulation of a relatively short duration of the actual process time of a complex AM process, for which both thermal and electromagnetic analysis is required; a smart coupling is used for decreasing the computational time, which still is in the scale that would not permit a simulation close to the real time of the process. In Figure 16, temperature plots of a two layer analysis that has been performed in [118] are depicted.
A different numerical approach of a thermal simulation is presented in [70], in which the thermal history of a component manufactured in the SLM process is modelled. The differential equation of heat conduction is discretized using finite differences and it is solved implicitly. The convection boundary condition is used for the surfaces of the part and a Gaussian profile is used for the laser heating. The moving of the laser heat source is simulated by changing the heating boundary condition over time. Also, the melting phase is taken into account, using the apparent heat capacity method and the material thermal properties are functions of both temperature and porosity. The analysis takes place in 2D, however, a calibration strategy of the model is presented for thin walled 3D parts. Finally, an algorithm which combines node birth and node distance adaptation over time is utilized in order to simulate the addition of new layers of while minimizing the computational time and cost, leading to upto 25 times faster computational times in comparison to standard node birth implementations. The model is capable of providing the temperature profile of the part during the whole manufacturing process, as well as the dimensions of the meltpool, even when using a standard personal computer. This model can be utilized for process optimization and, with further optimization, for process control as well. In Figure 17, a plot of the simulation of the temperature profile and the melt pool dimensions on the crosssection of the melt pool are depicted.
Fig. 14
Schematic diagram of the powder stream from a coaxial laser deposition nozzle, in which the powder consolidation plane is visisble [114]. 
Fig. 16
Temperature filed during a two layer deposition simulation (a) no induction heating, (b) induction preheating, (C) induction post heating [118]. 
4 Discussion
In Section 2, modelling studies have been presented for each AM process along with the process parameters used and the modelled KPIs. In order to summarize the presented studies, the different KPIs have been classified in groups and the following table has been created. It has to be noted that the classification into KPI groups is indicative and that some groups are very closely connected to each other, rendering difficult the choice of a category for the study of modelling. The KPI groups can be seen in the horizontal axis of Table 13, whereas the AM process groups are situated in the vertical axis. Also, a color coding has been used in order to indicate the modelling approach that has been followed in each work (black: analytical, red: numerical, blue: empirical).
More specifically, in this table the state of the art on modelling of AM processes that has been presented in this paper is summarized and classified according to (i) AM process, (ii) Modelled KPI, (iii) Modelling approach. The data of Table 13 have also been graphically presented in Figure 18, in which the percentages of studies that model the different KPI Groups of each AM process group are depicted and in Figure 19, in which emphasis is given in the number of studies that follow a certain modelling approach for the modelling of the different KPI groups.
It can be observed that in each process group different KPIs are those to which the majority of studies refer to. It can be said that, for that particular process group, those KPIs are more significant than the others. In VP, most of the studies model the mechanical properties and microstructure KPIs. In SLS most models refer to heat transfer related KPIs/phenomena, as well as in SLM and EBM, whereas in DED they are one of the two most important categories. In the aforementioned process groups, important also are the dimensional accuracy and microstructure KPIs, which are directly connected to thermal phenomena, as has been previously stated. Also, in the DED process, the droplet kinematics KPI plays a major role, whereas in MJ it is the most modelled one. In ME and UC, the mechanical properties and microstructure KPIs are the ones to which most of the studies refer to. Finally, in BJ the topology/dimensional accuracy and surface roughness share most of the research interest, as well as in LOM, in which surface roughness, mechanical properties and microstructure and heat transfer related KPIs/phenomena are the three most modelled KPIs. The fact that in each process group the KPIs that attract the highest research interest are different is due to the high importance of that particular KPI for that process group. More specifically, this KPI can be crucial to the process mechanism, encompassing the drawbacks of the process and a model capable of accurately simulating it, can be utilized for the improvement of the important drawbacks of each process group.
An observation concerning the strengths and weaknesses of the different modelling approaches has also been made. More specifically, analytical approaches are fast to run and provide an overview of the physics of the process, however, they tend to require more assumptions and simplification of the process. They are ideal for the modelling of KPIs, like build time and energy consumption, which are easily analytically calculated.
On the other hand, the use of empirical methods requires minimal simplifications and assumptions and leads to direct and validated conclusions. However, there is little to no connection to the physics of the process and the conclusions that are drawn from such models cannot be generalized, since they are directly dependent on the specific conditions of the model calibration experiments that were conducted. As a result, using such methods can be very practical for the solution of a specific problem, but they are unsuitable for the extraction of generalized conclusions concerning the identification and optimization of the problems faced by AM technology. Finally, the use of the numerical approach requires less assumptions than the analytical one, which leads to more realistic models, while it simultaneously provides an overview of the physics of the process. Moreover, models that use this approach are capable of describing not only the initial and final state of the simulated KPI, but also the progression of its values over time, the knowledge of which is of crucial importance for the problem detection and optimization of the process. However, the fact that the phenomena that take place in AM are highly dynamic both spatially and temporally lead to restrictions concerning the time span and part dimensions that can be modelled when the numerical approach is used, because the computational cost and time needed is extremely high.
Observing Figure 19, it can be seen that a certain modelling approach is preferred by the majority of the studies of each KPI group. This can be attributed to the fact that the different phenomena are more easily described utilizing a certain approach, whereas it is difficult to use the other approaches due to the nature of the KPI. As a result, a connection can be identified between the KPI groups and the modelling approaches: since some KPIs are more easily modelled using a certain approach, the strengths and weaknesses of the modelling approach also characterize the modelling of that KPI as well. More specifically, one of the major drawbacks of numerical modelling is the need of high computational power and of long computational times. Since the KPI groups of droplet kinematics, mechanical properties and microstructure and topology/dimensional accuracy and heat transfer related KPIs are better suited to be modelled using numerical methods, as can be observed in Figure 19, the modelling of those KPI groups faces the same drawbacks to those of the modelling approach that is best suited for the modelling of that group. However, since each approach has different strengths and weaknesses, it is important that modelling studies which utilize different approaches to be available for all the KPI groups and AM processes. This will lead to models of the same KPI that will, however, have different strong and weak points and as a result can be used in conjunction with each other, so as to minimize the weaknesses and maximize the advantages of such an approach. In this scope, Table 13 can be utilized for the identification of those gaps in literature, in order to be addressed by future studies.
KPI groups (horizontal axis) of the different AM process groups (vertical axis) modelled using analytical (black), numerical (red) and empirical (blue) approach.
Fig. 18
Percentages of studies that model the different KPI Groups of each AM process group. 
Fig. 19
Number of studies that follow a certain modelling approach for the modelling of the different KPI Groups. 
5 Outlook
In this study the existing literature on process modelling of AM processes has been presented and classified according to the (i) AM process, (ii) Modelled KPI, (iii) Process parameters used and (iv) Modelling approach followed; also, the gaps in the existing literature concerning the modelling approach used for the different KPI groups and AM processes have been depicted in Table 13 and in Figures 18 and 19.
Moreover, the following conclusion has been drawn concerning the suggested scope future modelling works should follow, so as to better cope with and help in the solution of the problems that are faced by AM today. More specifically, since most of the AM processes utilize heating as a form of bonding of new layers, the modelling of heat transfer related KPIs/phenomena is of crucial importance, because it directly affects the dimensional accuracy, topology, mechanical properties and microstructure of parts. Those are the KPI groups that mainly need to be improved in order to increase the quality of parts manufactured using AM. The complexity and coupling of those KPI groups constitutes the numerical approach as the most suitable one for their modelling, because it needs less assumptions than analytical methods, while it provides an overview of the physics of the whole duration of the process, including the transitional states which are of crucial importance. Also, significant is the fact that they are not dependent to any the specific experimental conditions, like empirical models do, and as a result they can be utilized as tools for the optimization of the process itself. However, the fact that the thermal phenomena that take place in AM processes are highly dynamic in time and space leads to restrictions concerning the time span and part dimensions that can be modelled. As a result, most of the existing modelling approaches are capable of simulating only a short time span of the manufacturing process, or a small section of a part due to the computational cost and time needed for such simulations. However, the knowledge of the entire progression of the phenomenon (thermal history, thermal stresses, thermal distortions) will enable the best possible optimization of the KPIs of the process, as well as provide a tool that can also be used for the optimization of the path planning of the heat source, which will lead to further improvements in the part quality.
Consequently, in order to address some of the most important issues hindering AM today, smarter numerical models, capable of providing accurate calculations of the thermal history and the oversizing of the printed parts, while minimizing the necessary computational costs (time and memory), have to be created. Such models will enable the selection of the process parameters that will maximize the quality of the produced parts, along with providing the necessary information for the optimization of the processes themselves.
5.1 Implications and Influences
As a review, this study identifies the most important problems of each AM process group and covers a wide range of modelling approaches, which aim at addressing them. In this regard, it can form a helpful reference, which can be quickly consulted, due to the simple tablebased structure classification that has been used. More specifically, it (i) provides important feedback to problems and challenges that have been addressed so far in AM, (ii) highlights literature gaps and (iii) suggests approaches for future modelling studies.
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Cite this article as: Panagiotis Stavropoulos, Panagis Foteinopoulos, Modelling of additive manufacturing processes: a review and classification, Manufacturing Rev. 5, 2 (2018)
All Tables
KPI groups (horizontal axis) of the different AM process groups (vertical axis) modelled using analytical (black), numerical (red) and empirical (blue) approach.
All Figures
Fig. 1
Issues of AM. 

In the text 
Fig. 2
Left: Free body diagrams of different types of motions of the powder particles [48], Right: Results of the model of [48]. 

In the text 
Fig. 3
Microstructure of the parts of the experiments conducted in [167]. 

In the text 
Fig. 4
Tensile strength specimens with various raster (−0.003 inch air gap) compared with injection moulded parts [137]. 

In the text 
Fig. 5
Clad geometric characteristics [82]. 

In the text 
Fig. 6
Clad formation. a) First stage: liquid clad, solid substrate. b) Final stage: liquid clad and substrate. 

In the text 
Fig. 7
Description and symbolism of the analytical model of Gharbi et al. [78]. 

In the text 
Fig. 8
Thermal fluid flow prediction of two different powder particles sizes of [56]. 

In the text 
Fig. 9
Error estimation percentages between the models developed in [44,177] and [45] respectively. 

In the text 
Fig. 10
Dimensionless solid layer thickness as a function of contact resistance [102]. 

In the text 
Fig. 11
Thermal contact resistance as a function of surface roughness [102]. 

In the text 
Fig. 12
Transition temperature as a function of impact velocity. Aluminum substrate with impact velocity of V_{0}, R_{c} contact resistance at the droplet–substrate interface. 

In the text 
Fig. 13
Photographs and simulation results of the numerical model of the study [104]. Numbers at the right indicate milliseconds following impact. 

In the text 
Fig. 14
Schematic diagram of the powder stream from a coaxial laser deposition nozzle, in which the powder consolidation plane is visisble [114]. 

In the text 
Fig. 15
Validation of the analytical model presented in [114]. 

In the text 
Fig. 16
Temperature filed during a two layer deposition simulation (a) no induction heating, (b) induction preheating, (C) induction post heating [118]. 

In the text 
Fig. 17
Temperature profile and melt pool dimensions on the crosssection of the melt pool [70]. 

In the text 
Fig. 18
Percentages of studies that model the different KPI Groups of each AM process group. 

In the text 
Fig. 19
Number of studies that follow a certain modelling approach for the modelling of the different KPI Groups. 

In the text 
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