The precise prediction of the clad geometry as a function of the employed process-  ing parameter per each used material is fundamental for additive manufacturing applications such as 3D printing. Many studies report the correlation between the cladding and the processing parameters (Chan et al. 1999). In Narang et al. (2012),  the outputs from the welding, such as the weld macrostructure characteristics, were mathematically modeled concerning the input process variables. Based on the weldment characteristics, including that of the bead contact angle, a mapping technique was developed for the graphical representation of the macrostructure zones’ shape profiles. Taking into account the complexity of the laser cladding process governed by the heat transfer among the laser beam, the substrate and powder, and mass transfer between the powder flow and the molten surface, many authors suggest that the best approach should be based on “combined parameters” (Felde et al.  2002).  This phenomenological approach uses simple mathematical formulae,  derived from a statistical analysis of measured data, to relate the laser cladding parameters with the geometric features of the clad track. Given the required clad height and available laser beam power, the proposed method allows one to calculate values of the scanning speed and powder feed rate, which are used to obtain low dilution, pore-free coatings, and fusion bonded to the substrate. Different sets of combined parameters are proposed to predict the track width and height for the coaxial and lateral geometry of the powder feeder (De Hosson et al. 2009). In Toyserkani et al.  (2003), the proposed model can predict clad geometry as a function of time and process parameters including beam velocity, laser power, powder jet geometry, laser pulse shaping, and material properties. Suryakumar et al. (2011) modeled the for- mations of single beads and overlapping multiple beads. While the individual bead’s geometry is influenced by the size of the filler wire and the speeds of the wire and torch, the step-over increment between the consecutive beads additionally comes into the picture for the multiple bead deposition. So, the geometry prediction is possible over many process windows for both powders and wire-based laser cladding techniques. In Hoadley and Rappaz (1992), a two-dimensional (2D) finite element model is presented for laser cladding by powder injection. The model simulates the quasi-steady temperature field for the longitudinal section of a clad track. It takes into account the melting of the powder in the liquid pool and the liquid/gas-free surface shape and position, which must conform to the thermal field to obtain a self-consistent solution. The model shows the linear relationship between the laser power, the processing velocity, and the thickness of the deposited layer. Another simplified model was proposed by Picasso et al. (1994). For a given laser power, beam radius, powder jet geometry, and clad height, this model evaluates two other processing parameters, namely, the laser-beam velocity and the powder feed rate. It considers the interactions between the powder particles, the laser beam, and the molten pool. In Naveed Ahsan and Pinkerton (2011) a coupled analytical–numerical solution is presented. Submodels of the powder stream, quasi-stationary conduction in the substrate, and powder assimilation into the area of the substrate above the liquidus temperature are combined. An iterative feedback loop is used to ensure mass and energy balances are maintained at the melt pool. The knowledge of temperature, velocity, and composition distribution history is essential for a better understanding of the process and subsequent microstructure evolution and properties. Numerical simulation not only helps to understand the complex physical phenomena and underlying principles involved in this process but it can also be used in the process prediction and system control. The double-track coaxial laser cladding with H13 tool steel powder injection is simulated using a comprehensive three-dimensional model, based on the mass, momentum, energy conservation, and solute transport equation. Some important physical phenomena, such as heat transfer, phase changes, mass addition, and fluid flow, are taken into account in the calculation. The physical properties of a mixture of solid and liquid phases are defined by treating it as a continuum media. The velocity of the laser beam during the transition between two tracks is considered. The evolution of temperature and composition of different monitoring locations is simulated (He et al. 2009). Bax et al. (2018) offer guidelines to evaluate process parameter maps for single tracks, which are a requirement for high-quality claddings and 3D structures. The procedure is executed by creating a process map for the parameters laser power, powder feed rate, and scanning speed. The relationship between process parameters and output responses and the interaction among the process parameters are analyzed and discussed in detail for Ti6Al4V. The analysis results indicate that powder feed rate is the dominant factor on the width and height of cladding coating while laser scanning speed has the strongest effect on the molten depth of the substrate (Sun and Hao 2012). The correlations that exist between key parameters of the process (i.e., laser power, scanning speed, powder feeding rate) and geometrical characteristics for single clads (i.e., height, width, dilution, and wetting angle) were predicted and analyzed by regression method (RA). The preliminary geometrical considerations allowed us to choose the processing parameters that led to high-quality clad with minimum porosity. All considerations finally resulted in the development of a processing map that shows the optimum parameters for the laser cladding process (Erfanmanesh et al. 2017). The influence of the laser power, scan speed, and laser beam focal position (focus, positive and negative defocus) on the shape factor, cladding-bead geometry, cladding-bead microstructure (including the presence of pores and cracks), and hardness has been evaluated. The correlation of these process parameters and their influence on the properties and, ultimately, on the feasibility of the cladding process, is demonstrated. The importance of focal position is demonstrated. The different energy distribution of the laser beam cross-section in the focus plane or in the positive and negative defocus plane affects the cladding-bead properties  (Riquelme et al.  2016). Ti–6Al–4 V deposits with variable thickness are made to assess the use of laser cladding as a repair technology. Both the effect of the building strategy (BS) and the incident energy (IE) on the metallurgical characteristics of the deposits about their complex thermal history have been studied. It is shown that for the configuration consisting of a decreasing track length (DTL) under high IE, a gradient of cooling rate exists that leads to the presence of different phases within the micro-structure. Conversely, homogeneous microstructures are present either for the con- figuration with a constant track length (CTL) under high IE, or for the strategy obtained from a DTL under low IE (Paydas et al. 2015). Francis (2017) analyzed three different process parameters, power, velocity, and spot size, on melt pool geometry for the electron beam wire feed and laser powder feed processes of Ti6Al4V. Beam spot size has been identified as having a major influence on melt pool geometry. It was also shown that experimental melt pool dimensions can be used to estimate how to spot size changes with a focusing parameter in additive manufacturing processes. Increases in spot size have been shown to eliminate the presence of keyholing, and a normalized spot size threshold is proposed to prevent keyholing in fave alloys.

Many experimental evidences demonstrate how the cladding properties depend on the substrate material. In Kumar and Roy (2009), a three-dimensional conduction heat transfer model is developed to predict the clad geometry (e.g., height, width, and dilution) and microstructure (scale and morphology) of the solidified layer for a laser cladding process. The effect of controllable input process parameters like absorbed laser power, powder deposition rate, and processing speed on the clad characteristics is critically assessed with the help of dimensionless parameters. A process map is developed that enables operators to pick up the proper process parameters for a feasible laser cladding process with desirable characteristics. Li and Ma (1997) found that the surface roughness (turbulence) of an overlapped clad- ding layer decreased with the increase of the overlapping ratio in an oscillating manner. At some overlapping ratios, the turbulence was at a minimum, and at some other ratios, it was at a maximum. Among various single-track sections, overlapping with symmetrical parabolic section single-clad tracks produced the smoothest clad- ding layer, in which the surface turbulence decreased in an oscillating manner. Lalas et al. (2007) have taken into account the processing speed and feed rate of the powder being supplied for the estimation of clad geometry. The surface tension between the added material and the substrate is used primarily for the calculation of the clad characteristics.

To obtain the powder packing information in the powder bed, dynamic discrete element modeling (DEM) was used (Lee 2015). The results show that negatively skewed particle size distribution, faster scanning speed, low power, and low packing density worsen the surface finish quality and promote the formation of balling defects.

In El Cheikh et al. (2012), a mathematical model implemented in the software Mathematica 8© is used to predict the clad cross-section dimensions and obtain an analytical description of the clad geometry. It was experimentally noticed that the cross-section shape is a disk due to the surface tension forces. Analytical relationships are established between the radius and the center of the disk on the one hand and the process parameters on the other hand.

During layer additive manufacturing, the cross-sectional profile of a single weld bead as well as overlapping parameters is critical for improving the surface quality, dimensional accuracy, and mechanical performance. Xiong et al. (2013) highlight an experimental study carried out to determine the optimal model of the bead cross-section profile fitted with circular arc, parabola, and cosine function, by comparing the actual area of the bead section with the predicted areas of the three models. A necessary condition for the overlapping of adjacent beads is proposed. The results show that different models for the single bead section profile result in different center distances and surface qualities of adjacent beads. The optimal model for the bead section profile has an important bearing on the ratio of wire feed rate to welding speed.

Laser cladding using scanning optics is a relatively little-studied matter. Scanning optics makes the adjustment of laser beam interaction zone numerically possible and it is, therefore, a more flexible optical tool than conventional static optics. A series of cladding tests were conducted using a 5 kW fiber laser and an oscillating linear scanner with dynamic powder feeding to determine the process characteristics and their possibilities and limitations (Pekkarinen et al. 2012). It was noticed that by using scanning optics, it is possible to vary the width and thickness of clad beads on a large scale. With scanning optics, it is possible to affect clad bead geometry so that only a 20% overlapping ratio is used. However, certain cladding param- enter combinations expose the clad bead to cladding defects. Also, a fast-moving scanned laser beam causes a wave formation in the melt pool that further causes stirring in the melt pool. The dilution was increased with an increase in the cladding speed.  However,  the increase of the dilution was dependent on the scanning amplitude.

Modification of the cladding angle during overlap has been observed experimentally and linked with the formation of an inter-run porosity. There is another group of models of the laser cladding process that try to model all physically involved processes, involving some approximation in equations, and solve these numerically. Physical phenomena including heat transfer,  melting,  and solidification phase changes, mass addition, and fluid flow in the melt pool, were modeled in a self-consistent manner. Interactions between the laser beam and the coaxial powder flow, including the attenuation of beam intensity and temperature rise of powder particles before reaching the melt pool were modeled with a simple heat balance equation. The level-set method was implemented to track the free surface movement of the melt pool, in a continuous laser cladding process. The governing equations were discretized using the finite volume approach. Temperature and fluid velocity were solved in a coupled manner (Qi et al. 2006). Laser beam cladding of metals by single-step powder delivery is analyzed with a process model that is based on balance equations of energy and mass. Effects like powder heating, clad layer formation, substrate dilution, and overlapping of tracks are discussed in dependence on the process parameters. In particular, the powder catchment efficiency and the beam energy redistribution in the material can be optimized by the powder mass flow rate and by the geometrical properties of the beam and of the powder jet (Kaplan and Groboth 2001). A three-dimensional finite element model is presented for precisely simulating the laser cladding process with a focus on dilution control (Zhao et al. 2003). Dilution is referred to as an important quality index in the laser cladding process, indicating the contamination level of the properties of the clad layer by substrate metals. As regards a good quality laser clad layer, low dilution as well as the metallurgical bond of the interface are prerequisites, so dilution control is essential in the process.