Prediction of geometrical characteristics and process parameter optimization of laser deposition AISI 316 steel using fuzzy inference

Laser metal deposition (LMD) process is an additive manufacturing technique that has attracted the interest of the automotive and aerospace industries due to its ability to manufacture parts with complex geometries and different types of metallic materials. However, the structure of the deposited layers and the geometrical characteristics of the manufactured parts are influenced by the interaction among the deposition process parameters. In this paper, fuzzy inference (FIS) technique was used to develop two models for predicting the geometrical characteristics and for optimizing the LMD process parameters. LMD was performed using AISI 316 stainless steel powder and substrate. An experimental design, based on factorial analysis, was used to correlate the influence of selected deposition process parameters, laser power (Lp), powder flow (Pf) and focal length (Fl) with the process geometrical characteristics bead height (Bh), bead width (Bw), depth of penetration (Dp), dilution (d) and wetting angle (wa). The factors Lp and Fl were used with three operating levels each, and the factor Pf was used with two operating levels. An analysis of variance allowed identifying that the Pf affects the Bh, Bh/Bw ratio, d and wa, as well as the increase in Lp showed an increasing of the geometric characteristics Bw and Dp. The first FIS, for predicting the bead’s geometrical characteristics, presented high adequacy (error up to 8.43%) for assessing the experimental conditions. The second FIS showed through the output defuzzified index (ODI) measured the best possible process parameters interaction, given the studied operating conditions and the output variables assessed.


Introduction
Hybrid manufacturing technologies (HMT) have attracted increasing attention from industry and academia [1]. Newman et al. [2] declare that the suitability of the HMT for combining processes/machines with flexibility and agility in changing manufacturing processes has allowed better utilization of resources and materials and reduces the time-to-market of complex parts and components normally used in the automotive and aerospace industries.
Flynn et al. [3] showed that two additive processes have been predominantly adopted by the industrial sector for metallic 3D fabrication: laser powder bed fusion (L-PBF) and laser metal deposition (LMD). Cortina et al. [4] affirmed that selective laser melting (SLM) is a common technique related to L-PBF process. This technique is based on the predeposited powder selective melting by a thermal source that usually is a laser beam that generates a thin layer of melted material, and for guaranteeing the unvarying powder distribution, a levelling system (roller or re-coater blade) is used.
Herzog et al. [5] asserted that this is a cyclic process until the 3D part is built. An advantage of this technique is the reuse of non-melted material. This is possible because L-PBF process is performed on a build platform inside an enclosed build chamber filled with inert gas. Liverani et al. [6] studied the effect of SLM process parameters on microstructure, defect formation and mechanical properties of 316L austenitic stainless steel. The results identified the correlation between the process parameters, microstructure and mechanical properties of the analysed samples. Defects in the deposition process as gas pores, voids and binging appeared with 150 W of laser power. Thus, a study using higher laser powers could allow analysing the correlation between high energy density and the deposition defects evaluated.
Shipley et al. [7] studied the process parameter optimization to address the challenges of martensitic microstructures, porosity and residual stresses during addition of Ti-6Al-4V. In this study, concerning process optimization, the part density maximization was the primary studied goal due to the negative effect of pores on fracture and fatigue properties. Gu et al. [8] studied the powder flow behaviour on fluid thermodynamics and laser processability of Ni-based composites. They conclude that the powder particle size had significant effect on the surface quality. Larger powder particles caused porosity on the part surface. However, they only studied the powder particle behaviour on two deposited layers and highlighted that future studies could be focused on applying the physical model to analyse the interaction of laser power and particle sizes and their effects on multi-deposited layers. Murkute et al. [9] studied 316L stainless steel clads corrosion-resistant deposited on carbon steel substrate using L-PBF-SLM process. They found that a higher energy density was necessary to produce low-defect dense clads with good adherence to the substrate. The porosity measurement and the relationship between projected layer height and penetration versus the ones achieved were not considered in the analysis.
The studies discussed above reveal interesting applications of L-PBF additive process and future directions to be explored. However, Cortina et al. [4] alleged that the main application of L-PBF processes has been the fabrication of full 3D parts with high-complexity shapes. Morgan et al. [10] sustained that two principal constraints associated to L-PBF process are the build rate and the size of the build chamber that limits the part size.
Flynn et al. [3] affirmed that the most common process integrated to a hybrid system is the laser metal deposition (LMD) technique, linked to directed energy deposition (DED) process, which is faster than L-PBF process. LMD is also known as laser engineered net shaping (LENS), direct metal deposition (DMD), laser cladding (LC) or direct laser deposition (DLD) [11].
In the LMD process, as explained by Schmidt et al. [12], the metallic powder is carried through a feed nozzle whereby a flow of inert gas (helium or argon) creates a protective area for depositing material. The powder is completely fused by a high-power laser beam in a layer-by-layer pattern to create 3D parts. Cortina et al. [4] indicated that LMD process has been advantageous due to its versatility for remanufacturing, building and repairing complex geometries and high valueadded parts. Other highlights have been the small heat zone generated during the deposition process and the higher deposition rate when compared to other additive processes such as SLM. They also suggest that, regardless of the application field of this additive process, there is an appealed necessity to develop predictive models to analyse the process parameter performance (depositing conditions) and response variables (geometrical characteristics) to evaluate, reduce and eliminate the deposition defects.
Several studies can be found in the literature exploring the development of mostly statistical models for simulating and predicting LMD process parameter behaviour assessing different response variables. Table 1 summarizes the key features of some of those studies, which are discussed next, including materials, experimental approach, key process parameters and output variables assessed.
Lee [13] used the Taguchi method (L18 orthogonal array) to maximize the deposition efficiency of cobalt powder alloy. The main conclusion was that the powder feed position had the greatest effect on deposition efficiency. Positioning the feed angle at 50°and the powder feed rate (0.62 g/s) as well as the laser travel speed (6.7 mm/s) had significant effects on deposition efficiency. The type of shielding gas (argon), the shape of the laser pulse (1.7/60 ms/Hz) and the focal position (10 mm) had virtually no effects. These characteristics provided a deposition efficiency 12% higher.
Sun and Hao [14] employed CCD (central composite design) and RSM (response surface method) to develop a mathematical model to analyse the influence of laser power, laser advancement speed and powder flow on Ti6Al4V (powder and substrate) deposition geometry, but without assessing hardness. The ANOVA method was used to establish relationships between process parameters and outputs. This ANOVA indicated that higher laser power (W) influences positively on the track width (mm). The powder flow (rev/min) had a significant effect on the coating width (mm) and height (mm), while lower laser scanning speed (mm/min) had a significant positive effect on the penetration depth (mm). The relative percentage error of the model reached 7%.
Graf et al. [15] performed a factorial analysis to determine the effect of process parameters on the cord geometry. The authors concluded that bead width (mm) is mostly influenced by the laser power (W), while the powder flow (g/min) has no effect at least in the range values they used. Bead height (mm) is mainly influenced by deposition velocity (mm/min) and powder flow. The procedure presented a good correlation between the statistical model and the experimental results obtained; residual values were lower than 0.1 mm.
Farahmand and Kovacevic [16] used a predictive approach to assess the influence of input parameters on geometry and microhardness of the deposited beads, established the dependency relationships between the inputs and outputs and optimized the deposition process parameters of AISI H13. They concluded that when increasing the laser scanning speed (mm/s), the bead height and HAZ (heat affected zone) decreased, while microhardness increased. The powder stream (g/s) had a positive effect on bead height and the laser power (W) on the bead width. The model relative error reached 5%.
Liu and Kovacevic [17] studied the effects of process parameters used on powder capture efficiency and layer geometry. They also used CCD/RSM/ANOVA to identify statistical relationships between input parameters and output variables and optimal processing parameters identification. This study used similar approach of previous works (Farahmand and Kovacevic, 2014) but using different powder materials, which could be considered an extended research. They concluded that the powder flow (g/min), the gas flow rate (SCFH) and the interaction between the gas flow rate and the standoff distance (mm) were the most significant factors affecting the bead height (mm), while the laser power (W) was the factor most affected the bead width (mm). The relative error reached 12% (error of the three predicted outputs). Yu et al. [18] studied through orthogonal experiments, with different deposition strategies, the process parameter optimization and performance testing of directed laser fabrication, seeking the improvement of mechanical and microstructural properties. The obtained results showed that laser power (W), scan speed (mm/s) and powder feed rate (g/min) affect the bead geometry and consequently affect the cladding samples' quality. The authors indicated that the depth of penetration (mm) and the penetration/ height ratio increased proportional to the calorific value (J/g). Higher energy values consequently have a lower powder feed rate (g/mm) or higher laser speed (mm/s), which affected the width (mm) of the deposition.
Wu et al. [19] developed a laser powder coupling numerical model with discontinuous powder feed rate for simulating the laser attenuation and heat transfer during laser powder coupling process. Gaussian beam (GB) profile with powder coupling was simulated on laser attenuation and heat transport. A super GB performance simulation was compared with ideal performance and presented a good correlation. However, the mean particle intensity under these two conditions varied 50%. A maximum error of 11% was reached.
Huang et al. [20] studied the process parameters correlation with the thermal characteristics for microstructure rapid prediction of laser powder-fed additive manufacturing (LPFAM). An analytical thermal model that couples the powder mass flow (g/min) and laser heat flux was used for analysing the thermal characteristics and the solidification parameters. The solidification parameters were combined with the substructure scale solutions for microstructure prediction, and finally the experiments of LPFAM were conducted to validate the correlation. The predicted microstructure showed reasonable agreement with the experimental observations (error was not CCD-RSM-ANOVA Laser power, scanning speed, powder feed rate Single-tracks geometrical characteristics, powder flow efficiency, microstructure quantified) for deposited material under different scanning speeds (mm/min). The authors declared that combining higher scanning speeds with a lower laser power (W) would result in lower energy input (J/mm 2 ); thus, a better microstructure was achieved. However, these characteristics make the deposition height (mm) smaller, which is not appropriate when a higher height for 3D manufacturing is expected. Ansari et al. [21] studied the laser water-atomized iron powder deposition. The experimental design and model for optimization model were implemented based on the RSM, and ANOVA was used for testing the regression model. The statistical model was developed to identify the optimal processing conditions to obtain just single tracks with minimum dilution, maximum bead height and high powder efficiency. Although the good correlation was achieved between experimental and predicted data (95% of confidence), the powder efficiency appeared low (20-40%) due to the irregular powder particle shape. The parameter interactions at the evaluated levels, energy density per unit area (J/mm) and powder flow rate per unit area (mg/mm) had a greater impact on the layer height (maximum).
As may be noted, the process parameters studied the most by researchers include, but are not limited to, laser power, laser scanning speed, powder flow, gas feed rate and laser beam diameter. Most of the researches explore only through statistical analysis, the effect of process parameters on the geometrical characteristics of the deposited beads. Hence, the Taguchi method, RSM, CCD, ANOVA and numerical models are the most studied.
The study of the deposition geometry not only implies the analysis of the bead height, width and depth of penetration as the main characteristics of deposition quality when assessing the fabrication of three-dimensional objects [22]. Likewise, other important geometrical features that have been studied in lesser extent are bead height/bead width ratio, wetting angle and dilution, which represent an opportunity to be explored in this paper.
The studies also focused on the statistical analysis of process parameter behaviour and optimization after retroactively analyse the geometrical characteristics of deposited materials. However, as part of the hybrid manufacturing process planning, it is of paramount importance to achieve a complete understanding of the correlation between input parameters and the response variable performance. Configuring models based on reliable computational tools to predict the part geometry assessing the critical attributes which affect the part quality can contribute to control and guarantee the process stability to prevent the appearance of deviations between the designed and manufactured part (target product profile).
Some researchers have applied fuzzy logic to evaluate parameters and manufacturing processes performance. Supriadi and Manabe [23] used an adaptive fuzzy controller to enhance the dimensional accuracy of dieless tube-drawing process. They found that with the introduction of the fuzzy approach, the process error variation was minimum. However, a more in-depth analysis is necessary, of how to avoid the process delay which consequently increases with increasing feeding speed, causing large errors at the beginning of the drawing process. Lin and Lin [24] used grey-fuzzy logic based on orthogonal array for optimizing the electrical discharge machining (EDM) process with multi-response. The experimental results demonstrated the effectiveness of the approach used.
Wu and Wang [25] presented a neuro-fuzzy approach for generating mould/die polishing sequences. The results experimentally obtained agreed with the predicted. Nevertheless, a source of error was possibly introduced due to the uncertainties of the surface roughness measurement since the measuring device used in the experiments had a limited measurable precision. Nagesh and Datta [22] used an artificial neural network (ANN) to predict bead geometry and penetration in shielded metal-arc welding. The model proposed presented high reliability on the evaluation of processes and parameters analysed, but the experiment results showed a total error of 20% of difference between estimated and experimental values.
Recently, Tan et al. [26] presented a Mamdani-type fuzzy inference system to predict tensile properties of as-cast alloy. The approach used combined with genetic algorithm presented high accuracy. Genna et al. [27] developed a fuzzy decision-making approach to optimize laser-assisted joining process of polymer-metal hybrid structures also combined with genetic algorithm that allows to obtain the results with a lower uncertainty level. Tasé-Velázquez et al. [28] analysed and simulated the input energy density generated after depositing AISI 316 steel single beads by LMD using fuzzy inference, and the FIS modelled presented high adequacy assessing the experimental runs showing 4.76% of error.
It was noted that no works were reported for studying the development of fuzzy approaches for predicting the geometrical characteristics and optimizing laser AISI 316 deposition process parameters, which represents a research topic to be explored. As a preliminary study developed on the Hybrid additive-subtractive Manufacturing Process Planning (HMPP), this paper aims to evaluate and predict the AISI 316 single-bead geometrical characteristics, such as bead height (Bh), bead width (Bw) and penetration depth (dp), as well as the optimization of LMD parameters that attend the process stability and better geometrical characteristics. The predictive model was configured using fuzzy inference-based approach (FIS). The behaviour of input parameters and their effect on the bead height (Bh), bead width (Bw), penetration depth (dp), dilution (d), Bh/Bw ratio and wetting angle (wa) was assessed to correlate the responses with the configured predictive FIS.

Experimental procedure
In this work, a hybrid manufacturing system that integrates the additive (LMD) and subtractive (5-axis machining) processes is studied. However, the focus relied on the additive process as part of a preliminary study to understand how to ensure the stability of this manufacturing process. Qualitative and quantitative analysis of the effects of process input parameter variability (Lp, Fl and Pf) on the output variables assessed (Bh, Bw, dp, Bh/Bw ratio, d and wa) was performed. The ANOVA was used to determine the statistical correlation and influence of input parameters on output variables. The fuzzy inference modelling for Bh, Bw and dp prediction and process parameter optimization was shaped on MATLAB software through the fuzzy toolbox. In addition, 2D and 3D surface graphs were plotted to complement the analysis.

Design of experiment (DoE)
The experimental design allowed the identification of process input parameters and their operating levels that directly affect the deposition quality. A factorial analysis was performed to determine all possible parameter combinations to improve the experimental effort and assess its interaction effects on the analysed response variables Bh, Bw, dp, Bh/Bw ratio, d and wa. The factors combination and their operating levels implied that 18 experiments (Runs) were performed. For each run (R1), two other replications (R2 and R3) were conducted totalling 54 experiments (three for each parameters combination) aiming at maximizing the statistical analysis' reliability for each run and correlating with the fuzzy modelling configured. Table 2 presents the studied process parameters and their operating levels.

Hybrid system and sample characteristics
The experiments were conducted in a ROMI's full 5-axis DCM 620-5X Hybrid Machine equipped with an AMBIT ™ deposition head with a 500 W maximum ytterbium fibre laser, with 1070 nm wavelength and 1 mm of maximum spot diameter. Figure 1 shows the configuration of the hybrid system and its operational axes, including the laser, shielding gas and powder feed operating principle regarding the substrate position.
The laser beam was focused on the substrate material (AISI 316 SS) to generate eight single beads (B 1 to B 8 ) on 18 samples, combining the process parameters and their different levels ( Table 2). The dimension of the samples was Ø 25×10 mm. Before performing the additive operation, the samples were face-milled 1 mm to produce a flatness surface for the deposition process resulting in Ø 25×9 mm as final sample dimension. The deposition strategy followed was a one-way pattern (Zig), and the powder used was AISI 316 SS in the range of 44-106 μm particle size. The length of each single bead was 10 mm, and the distance between the beads was 2 mm. The powder was fed simultaneously, by a coaxial hopper focused on the laser spot with 90°orientation relative to the substrate, with argon gas. Argon gas serves as a shield to protect deposited material and preserves it from oxidation [30].

Bead geometry analysis
The metallographic work was done manually. It includes the samples cross-section, the preparation on chemical reagents and sample polishing. The measurement of Bh, Bw and dp on the substrate (Fig. 3) was done using an optical microscope Nikon Optiphot, the lens, with 1 mm measurement pattern, inserted was ×100.
The measurements of the bead height (Bh), bead width (Bw) and depth of penetration (dp) were made along each bead of the cross-sectioned samples including the two replications. For example, considering the run 1 and two replications were generated 3 samples. Each sample was crosssectioned resulting in 6 halves, and each bead (8) of every half was measured as well. Therefore, the quantity of measurements for one output variable (ex. Bh) was 48×18 runs equal 864. An average value was calculated for each variable  The Bh/Bw index represents the convexity rate of the deposition. This index is discussed in Rao et al. [31]. They developed a model for pulsed gas metal arc welding (P-GMAW) to predict weld bead geometry. In this study, the-smaller-the better index implies a good geometrical characteristic. In our study, near 0.5 index is expected since it contributes to avoid overconvexed layers when overlapped. As indicated in Caiazzo [32], dilution (d) is a measure of the bead mixing with the substrate. It represents the percentage of deposition penetration in the substrate considering the entire bead height. A higher mixing will lead to better characteristics expected as it contributes to lack of fusion avoidance. Calleja et al. [33] indicated that for wetting angles (wa) outside the range of 0-80°, the layers overlapping was considered defective. This defect could be conditioned for voids appearance between them. Dilution (d) and wetting angle (wa) were calculated following Eq. 1 and Eq. 2, respectively, based on bead height (Bh), bead width (Bw) and depth of penetration (dp) average measurements. d % ð Þ ¼ dp Bh þ dp *100 ð1Þ

FIS for bead geometry prediction
Through fuzzy logic is possible to model complex processes that contain uncertain and vague information with less resources use in terms of hardware and software and has shown to be an effective technique for addressing linguistically specified goals [34]. Gomide et al. [35] indicated that the configuration of a fuzzy inference system (FIS) includes the following: (i) the input data, which are non-fuzzy sets of data previously defined by experts or resulting from measurements or observations; (ii) the fuzzification module, which transforms the non-fuzzy data into fuzzy sets [36] so they may become instances of input and output linguistic variables; (iii) the fuzzy reasoning module, that can be implemented commonly using the Mamdani approach [37], to determine how the rules base specified following the "if"-"then" constraints is activated for a given situation, and combined along with input fuzzy data, the module infers the control actions; and (iv) the defuzzification interface, which transforms the information correlated with the output linguistic variable into a numeric value that can be measured. This interface requires the definition of a defuzzification method [38], which allows obtaining a defuzzified output that is a logical response correlated to the system under study. The FIS configuration to predict the deposition geometrical characteristics, namely Bh, Bw and dp, is shown in Fig. 4. To evaluate the output variables (Bh, Bw, dp), a different fuzzy system was structured for each one.
The fuzzy sets, linguistic variables and membership functions generated for each input parameter are presented in Fig. 5, and the response variables are analysed in Fig. 6. The trapezoidal shape membership functions define the fuzzy sets for input parameters and response variables. Concerning the input linguistic variable "laser power" were defined four input membership functions and their input fuzzy sets, which were classified as low, medium, high and very high according to the parameter operating range. Regarding input linguistic variables "focal length" and "powder flow", three membership functions and their input fuzzy sets were defined and classified as small, medium and large. The boundaries of the membership functions corresponding to the input parameters (Fig. 5) were defined following the operating levels presented in Table 2.   For each output linguistic variables (Fig. 6), three membership functions and its output fuzzy sets were defined as low, medium and high, respectively. The boundaries of the membership functions corresponding to the output variables ( Fig.  6) were defined following the expected geometry considered suitable and its correlation to the dependent variables Bh/Bw index, dilution (d) and wetting angle (wa) as previously defined in Section 2.3.
The established rule base followed the If-then sequence for each input parameter condition (antecedent) and each response variable (consequent). A total of 108 rules were defined for the predictive fuzzy model that analysed the response variables (Bh, Bw and dp). For each variable analysis, 36 rules were configured for delimiting the rule base following the "if-then" constraints (i.e., if laser power is "high" and focal length is "medium" and powder flow is "High" then deposition height is "acceptable"). The correlation between variables and their membership functions follows an exponential configuration m n to define the number of rules, where "m" represents the quantity of membership functions and "n" the quantity of variables.
The fuzzy inference module was performed using the Mamdani (max-min) method, which performs a "fuzzy reasoning", from the rules defined to generate a fuzzy value. In the last stage of the system, the defuzzification module allows converting the fuzzy output to a crisp (measurable) output.
The defuzzification method used in this paper was the socalled centre of gravity (COG) method [39], as following the Eq. (3), which returns as measurable output of the centre of the inferred membership function area of the parameter analysed: where y 0 is the defuzzified output of the predicted response variables, y is the variable centre value of fuzzy regions and μ x (y) the aggregated membership functions. MATLAB offers the possibility to choose the defuzzification method, which facilitate the computational processing. In this article, the FIS predicted values of bead height (Bh), bead width (Bw) and depth of penetration (dp) for each deposition condition studied were called "output defuzzified measure" (ODM) and compared along with the experimental analysis.

FIS for process parameter optimization
A new fuzzy system (Fig. 7) was structured to determine which LMD process parameter operating levels simulate the best performance. The inputs of this system were defined from the response variables (Bh, Bw and dp) predicted from the previous fuzzy system (Fig. 4).  The greater height, width and depth of penetration, the better geometric feature is expected. Given the precept "higher-the-better", the ODM for each response variable was normalized by Eq. (4) as established in [40]: where * N i (k) is the normalized value and ODM i (k) is the output value for the analysed response k from experiment i. The same procedure of the previous system ( Fig. 4) was followed to model the optimizing fuzzy system of Fig. 7. The fuzzy sets, linguistic variables and membership functions of the system inputs and outputs (ODIoutput defuzzified index) are presented in Fig. 8. For input linguistic variable "bead height", three input membership functions and their input fuzzy sets were defined and classified as low, medium and high. Regarding the input linguistic variable "bead width", also three input membership functions and their input fuzzy sets were defined and classified in small, medium and suitable. Concerning the input linguistic variable "depth of penetration", also three input membership functions and their input fuzzy sets were defined and classified as unsuitable, acceptable and suitable. The ODI membership functions and their fuzzy sets were classified as low, medium and high, and the boundaries of the membership functions corresponding to the normalized values do Bh, Bw, Dp and estimated ODI followed the precept defined before ("higher-thebetter"), and normalized values using Eq. (4).
Twenty-seven (27) rules were structured for this new fuzzy system based on the precept that the better performance of process parameters will respond to a higher output value. The inference system used was Mamdani and the COG defuzzification method. The FIS output to optimize the process parameters was called "output defuzzified index" (ODI). This index behaves in the range from 0 to 1. The higher ODI value indicates that a given deposition condition is the better one that simulates the LMD process. Table 3 summarizes the measured mean values for bead height (Bh), bead width (Bw), depth of penetration (dp), bead height/bead width (Bh/Bw) ratio, dilution (d) and wetting angle (wa) for each deposition condition. Mean values were obtained from the measured values of the 18 experiments and its replications.

Effect of process parameter interaction on deposition geometry
From a separate evaluation of each response, it can be noted that in experiments performed with powder flow equal to 13.35 g/s, the Bh and Bw increases, and the depth of penetration decreases. Dilution (d) indicates the percentage of deposited powder's penetration in the substrate, the behaviour ranged from 8 to 22%. The highest d percentage was achieved at Lp = 275 W, 5 mm and 5.2 mm of focal length (Fl) and Pf = 9.40 g/s. The highest wa was obtained in experiment 2 with a value of 35°with a Lp = 225 W, Fl = 4.8 mm and Pf = 13.35 g/s. The greatest height was observed with Lp = 275 W, Fl = 5 mm and Pf = 13.35 g/s. The largest width was reached at Lp = 275 W, Fl = 4.8 mm and Pf = 13.35 g/s. Penetration (dp) was highest at Lp = 275 W, Fl = 5 mm and Pf = 9.40 g/s. The Bh/Bw indexes were measured between 0.339 and 0.698, varying 51% approximately, and the index values next to 0.5 were reached in the experiments 1, 7 and 13.  9 Mean values of Bh, Bw and dp for Lp (W) and Pf (g/s) interaction

Analysis of variance (ANOVA)
The ANOVA was used to evaluate which parameters have a significant influence on the process and on the evaluated variables. Tables 4, 5, 6, 7, 8 and 9 present this analysis. P values <0.05 indicate that the parameter effect on the evaluated output variables is significant at a 95% confidence level. Tables 4,7,8 and 9 show that the powder flow seems to have greater influence on the Bh, Bh/Bw ratio, d and wa. For Bw and dp, laser power seems to have a greater significance (Tables 5 and 6). Figure 9 illustrates the mean values of Bh, Bw and dp, and Fig. 10, the mean values of Bh/Bw ratio, dilution (d) and wetting angle (wa), for each interaction of the input parameters Lp (W) and Pf (g/s). Figure 11 shows eight cross-sectioned beads to illustrate the general variation trend on the deposited geometry (deposition initiating from left to right). It was noticed that in each sample, the first three beads had lower height but with greater depth of penetration, accounting for approximately 31-48% of penetration in relation to height. The opposite is true for the remaining five beads, showing a variation between 8 and 13%. This may be conditioned by the initial oscillation level of powder flow, which suffers a 12-s delay to be stabilized before initiating the material deposition.
The previous results analysed (Sections 3.1 and 3.2) are in line with the results showed in previous papers discussed [14; 15; 17; 18; 21; 31-33] 3.3 Fuzzy inference for Bh, Bw, dp prediction and LMD process parameter optimization Figure 12 presents the rule viewer toolbox of the ODM of Bh for the development of the predictive fuzzy inference system. The rules-viewer is used to determine each expected output according to each experimental condition performed. The first three columns show the three input process parameters, and the fourth column indicates the ODM of the response variable (bead height). Using experiment 18 (Table 3) as example, it can be noted that for Lp = 275 W, Fl = 5.2 mm and Pf = 13.35 g/s, the ODM for bead height is 0.413 mm. Comparing the model results with the experimental value obtained (0.416 Fig. 10 Mean values of Bh/Bw ratio, dilution (d) and wetting angle (wa) for Lp (W) and Pf (g/s) interaction Fig. 11 Cross-section of eight strands generated in the LMD process mm), the relative error is 0.72%, which can be considered low. The predicted Bw and dp values were also analysed from a rule viewer like Fig. 12. Although the focal length was considered part of the fuzzy model since it was settled for operating at three levels, the ANOVA proved that it had no influence on the responses evaluated. Figure 13 presents the 3D maps obtained during the predictive fuzzy modelling. Each map shows the interaction of the two process parameters (Lp and Pf) that has greater incidence on the outputs (Bh, Bw and dp) according to the ANOVA. Figure 13 shows the ODM 3D maps in (a) for predicted Bh, (b) for predicted Bw and (c) for predicted dp. The region with maximum predicted value for the evaluated output variables is represented with the strongest coloured surface. Figure 14 indicates rule viewer toolbox of the ODI obtained from the normalized values of height, width and depth of penetration as part of the fuzzy system for LMD process parameter optimization. As an example, experiment 18 was indicated again to exemplify that for height index 0.149, width index 0.282 and depth of penetration index 0.211, the output index was 0.171. The maximum ODI indicates that the fuzzy system predicts the LMD process parameters for that experiment has the best performance. Table 10 presents the results obtained from the fuzzy systems for prediction of bead height, width and penetration depth indicated in ODM columns, as well as, for simulation of the best performing LMD process parameters as shown in ODI column. The maximum ODI was 0.845, which corresponds to experiment 9, with 250 W of laser power, 5 mm of focal length and 9.40 g/s of powder flow.
The comparative evaluation between the results obtained from the predictive fuzzy inference system and the experimental data is presented in Fig. 15 for Bh, Bw and dp and Fig. 16 for, d, wa and Bh/Bw ratio. As the three last output variables are dependent from the ones simulated, the comparison between predicted and experimental is also presented. It can be seen that there is a good correlation between both results. The relative error (Table 10) calculated from the experimental data and the ODM indicates that the fuzzy model presents high accuracy for Bh, Bw and dp prediction. The total error for each predicted variable ranged from 1.03, 2.88 to 4.8%, while the average error was between 0.06 and 0.27%, which is very low.

Conclusions
In this paper, a technique of fuzzy inference was used to develop two models for predicting the geometrical characteristics (bead height, bead width and depth of penetration) and for optimizing the LMD process parameters using AISI 316 stainless steel powder and substrate. The main conclusions of the work allow us to affirm that: that the powder flow parameter had a significant effect on Bh, Bh/Bw ratio, d and wa, while, on Bw and dp, the laser power had the greatest influence. On Bw and dp, the laser power had the greatest influence. It was noticed that there is a proportional relationship between height/width index and the wetting angle. The greatest height (Bh) was 0.454 mm, and the largest width (Bw) was 0.743 mm. The highest depth of penetration (dp) was 0.069 mm.  Bold values represent that experiment 9 presents the optimized values for each parameter after run the fuzzy model for processs parameter optimization indicating the best performance in the material deposition & The results of the predictive fuzzy model presented a high correlation index when compared with the experimental data. The total error was up to 8.74% and the average error lower than 3%. The fuzzy system structured to determine the best interaction of process parameters that meets the fuzzy system optimization features indicated that the 0.845 ODI represents a laser power of 250 W, 5 mm of focal length and 9.40 g/s of powder flow. & The developed fuzzy model can be improved and integrated with an automated LMD process environment to reduce the hybrid process planning activities complexity and increase process stability. & Future research could focus on studying the influence of the variation of these parameters on the same variables studied and on the efficiency of multilayer deposition and with different types of powder. & The results show that the fuzzy inference model based on the Mamdani method provides an accurate prediction of the evaluated output variables and good reliability and stability of the approach.
Acknowledgements The authors would like to thank the Higher Education Personnel Improvement Coordination -Brazil (CAPES -Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) which financed this study in part -Finance Code 001. In addition, the authors would like to thank the ROMI Industry for allowing the open and wide access to their facilities for studying the hybrid system ROMI DCM 620-5X HYBRID.
Author contribution Daniel René Tasé Velázquez designed and implemented the fuzzy models and analysed the results. André Luís Helleno designed the experiments, helped verify the results and supervised the project. Hipólito Carvajal Fals helped to collect the data and measurements. Raphael Galdino dos Santos helped to perform the tests, collect data and verify the results. All authors discussed the results and contributed to the final manuscript.
Availability of data and materials The data that support the findings of this work are available on request from the corresponding author.

Declarations
Ethics approval Not applicable