The effect of process parameters on geometric deviations in 3D printing with fused deposition modelling

Despite the large diffusion of additive manufacturing, and markedly fused filament fabrication, some quality aspects of the 3D printed parts have not been dealt with sufficiently. This applies particularly to geometric accuracy and the influence process parameters have on it. The paper describes an experiment in which 27 copies of a part were manufactured by means of a desktop fused filament fabrication device while manipulating layer thickness, printing speed, and number of contours. The effect of such process parameters on five typologies of geometric deviations and the duration of the printing process was assessed. While all the process parameters showed effects on both the printing time and some geometric deviations, the number of contours resulted as the most critical factor. The paper includes a proposal to optimize geometric accuracy and the rapidity of the process, which foresees the maximization of the number of contours, the minimization of the layer thickness, and the use of an intermediate value for printing speed.


Introduction
Fused filament fabrication, commonly indicated with the commercial name fused deposition modelling (FDM), is an additive manufacturing (AM) technology used to print parts that are often characterized by geometrically complex shapes [1,2]. FDM is typically considered a reference process for rapid prototyping (RP) [3]. Nevertheless, this technology is evolving continuously and rapidly. FDM-printed parts are no longer used for esthetic prototypes and displays only, but also as end products. Examples include drilling grids in the aerospace industry [4], foot prostheses [5], and edentulous mandible trays [6] in the bio-medical field. It is evident that these products require specific performances. Performances to be attained typically concern mechanical properties, surface finish, defects, and other quality-related aspects. Much research considers a subset of these performances to limit the complexity of presented studies. Consistently, the authors dedicate here special attention to quality and, markedly, geometric accuracy, as explained and motivated in the following section.
Generally, to cope with quality requirements, the optimization of process parameters is of paramount importance. This is clearly not limited to FDM process [7,8], but it extends to other AM techniques, such as powder bed fusion [9,10], or wire and arc additive manufacturing [11,12]. The effects of process parameters have been investigated to identify the most impacting ones and their relations to performance measures [13][14][15][16][17][18][19][20][21]. Despite these numerous attempts, the production of functional parts with FDM is still limited due to the limited accuracy of prints if compared with traditional manufacturing process like turning and milling. Xinhua et al. [22] is a case in point in this regard.
The study of Mohamed OA et al. [23] is a first attempt to summarize the studied relations between process parameters and product performance measures in FDM, but other studies have followed its publication. Section 2 extends the review of the effect of process parameters on quality performances and the duration of the 3D-printing process, which is a plainly connected aspect. The extension of the review sheds light on the limited attention paid to geometric accuracy. This takes place despite geometric accuracy is expected to play an important role in objects created through FDM. On the one hand, FDM ambition to become a commonplace technology for functional products determines the need to cope with quality aspects that, inherently, include geometric accuracy. On the other hand, limiting geometric deviations can be seen as a prerogative for RP too. Geometric issues are supposed to affect esthetics [24] and perceived quality of parts [25], with a potential detrimental effect on design decisions following the RP stage.
The overall scope of the paper is therefore to acquire more knowledge about the role played by process parameters on geometric accuracy. As process duration is a known issue in AM (and even more markedly in FDM) and accuracy goals are typically offset by the extension of the printing time (PT), the latter is considered contextually in this study. The specific objective of the paper is then a proposed optimization of geometric accuracy and PT.
The residual of the paper is organized as follows. Section 2, beyond contributing to the understanding of the studied effects of process parameters on quality features and process duration, specifies the methodological objectives. In particular, manipulated process parameters and tested geometric deviations are selected. Section 3 illustrates the methodological procedure followed to obtain the data necessary to pursue the paper's objective. While results are presented and discussed in Sect. 4, conclusions are drawn in Sect. 5.

Background
By considering [23] as a starting point to identify pertinent contributions and common process parameters, the literature review was expanded to stress the attention paid to this topic. The focus of the authors' review included the typology of relations between process parameters and investigated affected factors. The outcome is a table ( Table 1) that shows the core relations between process parameters (first column), and product/process performances (first row). The definition of the former within FDM is taken for granted; readers can benefit from [26] for clarifications. The latter are defined with a positive acceptation to indicate the desirability of the effects, hence as follows: • Surface finish instead of roughness • Dimensional and geometric accuracy instead of deviations or errors • Process speed instead of PT.
The cells of Table 1 report the sources that have analyzed the relation between the former and the latter. Relations are classified as follows: • Positive (direct) if the increase of the process parameter results in a statistically significant improvement of the performance measure. This kind of relationship is marked by its arrangement in the respective sub-column featured by an upwards arrow. • Negative (inverse) if the increase of the process parameter results in a significant worsening of the performance measure. This kind of relationship is marked by its arrangement in the respective sub-column featured by a downwards arrow. • Neutral if the tested relationship was neither significantly positive nor negative. This kind of relationship is marked by its arrangement in the respective sub-column featured by a dash.
As readers can notice, studies have given rise to different, if not conflicting, results. This might take place because the relation between process parameters and accuracy is possibly mediated by materials, geometry, or other factors. For instance, a plain contradiction concerns the influence of raster width on surface finish. An increase of raster width (obtained through a variation of material flow rate) from 0.3 to 0.5 mm improves the surface finish of the printed piece [27]. Conversely, [28] states that by increasing the raster width (0.4 mm, 0.53 mm, 0.66 mm), the surface finish worsens.
The attention paid to geometric accuracy is hitherto limited. Among the few contributions, [35] shows that specific dimensional and geometric accuracy performances can be attained by acting on nozzle temperature, layer thickness (LT) and infill density. Similarly, [36] analyzes the geometric accuracy of circular features, concluding that different process parameters, such as raster angle and air gap, can strongly influence local dimensional deviations over circumferences and diameters. In [56], the scholars customized the design of a test sample to measure key features of the components quantifying the effect of some process parameters (e.g., LT, extruder temperature, and infill density) on the geometric accuracy resulting by the process. They finally found an optimal combination of these parameters. Another contribution was made by [59], which proposes a detailed analysis of the effects of layer cooling time on a part's geometrical deviations. The contribution further discusses the overall printing speed (PS) and issues concerning the extruder's temperature. Eventually, [60] compares printed and machined holes in specimens, investigating the effects of standard process parameters on dimensional and geometric deviations. [59] [50] Beyond the limited consideration of geometric accuracy in FDM studies, it is worth noting that available studies have failed to address to what extent the achievement of geometric quality can be counterbalanced by PT. In order to fill this gap, the authors have designed and conducted an experiment, as made apparent in Sect. 3.
The first step was to select some process parameters for the scopes of the experiment. With the aim to identify significant parameters, some were chosen from previous studies giving rise to no conflicting outputs and that could logically affect geometric properties of 3D-printed parts and, contextually, the duration of the printing process. The listed parameters below are characterized by a strongly uneven popularity (high, average, low) in previous studies.
• LT is the most studied parameter according to Table 1.
Its impact on performance measures and process speed is straightforward in many studies. LT is a good candidate for affecting geometric quality because of its recognized role in the staircase effect of 3D-printed parts, e.g., [65].
A few studies have already considered this parameter in relation to geometric accuracy, but PT was never considered contextually. • PS is possibly relevant when the accurate deposition of material is requested. The magnitude of its influence on process speed has been surprisingly overlooked based on Table 1. • Number of contours (NC) can be crucial when it comes to the accuracy of external surfaces. Its effect on process duration has emerged as marginal so far.
The geometric deviations that will be addressed in this paper are selected for convenience and reported in Table 2 along with their definition based on the standard 1SO 1101:2017 [66] about geometrical product specifications. Despite the peculiarities of AM technologies, the ISO 1101 has still to be considered when it comes to geometric dimensioning and tolerancing (GD&T) following ASME Y14.46 [67], as reported in [68,69].  Minimum (radial) distance between two coaxial cylinders defining a region of space which entirely contains the reference surface.

Concentricity
Minimum diameter of a cylinder having its axis parallel to a datum that defines a region of space which entirely contains the axis of reference cylindrical surface. Surface parallelism Minimum distance between two parallel planes (parallel to a datum) defining a region of space which entirely contains the reference surface. Surface perpendicularity Minimum distance between two parallel planes (perpendicular to a datum) defining a region of space which entirely contains the reference surface.

Materials and methods
In order to understand how the three selected parameters affect geometric deviations, a set of parts characterized by different process parameters was printed and hereafter measured with a coordinate measuring machine (CMM) through a developed automatic measuring procedure. The choice and design of the specimen are justified in the subsection that follows. Subsequently, the printing and measuring processes are described.

Design of the part
As no standard shapes are used in the study of geometric deviations for AM technologies, a part has been designed, which lent itself to the assessment of all the geometric deviations reported in Table 2. The part has been designed with the following features ( Fig. 1): • A cylinder at the base of the piece (C1). • Two concentric holes with a cylindrical shape and different diameters (C2 and C3) that are theoretically concentric even with C1. • Eight surfaces perpendicular to C1 (S1, S2, S3, S4, S5, S6, S7, S8) arranged as a regular octagon.
A complete draft of the part including dimensions can be found in Appendix Fig. 33.
In this way, it has been possible to obtain features with different orientations and thus mitigate any possible effect of the building orientation. Horizontal, vertical, and inclined surfaces have been conceptualized for both planar and cylindrical surfaces so to avoid the need of support material. Moreover, the horizontal surfaces have been conceived with different heights in order to attenuate any influence of the distance from the printing plane.
To avoid the use of support material, no horizontal holes were included in the part, and the inclination of surfaces I1-I8 and holes C4-C5 are limited to 30° from the vertical direction. Two inclined holes only were made in order not to reduce the surface area of the I1-I8 surfaces excessively and, therefore, not to compromise their measurement. The little "bulge" (feature D1) made on C1 near the printing plane is a reference that facilitates the consistency of the measurement operations by identifying the correct faces.

Printing process
In the present study, a single filament Ultimaker 2 + printer (located at the Bitz Fablab of the Free University of Bozen-Bolzano) was used (Fig. 2). The detailed printer specifications are shown in Table 3.
All parts were printed with an infill density of 20% with a grid pattern; the utilized material is a common white filament of PLA for a 0.4-mm diameter nozzle.
A set of control levels has been assigned to each of the three process parameters (i.e., LT, PS, and NC); the selected values for each parameter are reported in Table 4. The process parameters have been set to achieve a low, medium, and high value in accordance with the capabilities (and reasonable operating ranges) of the used 3D printer.
The number of objects to be printed was given by all the possible combinations considering the three levels and the three parameters. Therefore, in order to conduct a fully factorial experiment, the number of pieces to be printed was 3 3 = 27. Table 5 reports the different values for the parameters and the levels assigned. It is evident that every piece corresponds to a unique combination of the selected parameters. Table 5 includes the PT for each combination.

Measuring process
In order to detect geometric deviations among the 27 FDM-ed parts, they were measured by means of a CMM (Coord 3 Ares NT), located in the Mechanical Lab of the Free University of Bozen-Bolzano (Fig. 3). The machine is equipped with a PH10 5-axis touch-trigger system. The workstation is furnished with a desktop computer equipped with the Coord 3 metrology software TouchDMIS, utilized to digitally record measurements.
Furthermore, the measuring process was supported by a bespoke clamp designed for correct positioning of the   parts to be inspected. This clamp prevents undesired movements of the piece during the measuring process. Moreover, it eases the positioning of parts with consistent orientation (due to the bulge, as aforementioned). This results in speeding up the process of alignment and, therefore, starting the automatic measuring procedure. Pictures of the support clamp are included in Fig. 4.
After an initial alignment of the coordinate systems of the part and the touch probe by manual probing of plane P2, cylinder C2, and surface S3, the automatic procedure was set. This allows an automatic measuring mode to take place. The measuring procedure had to ensure appropriate coverage for all the surfaces to be measured. More in detail, the procedure foresaw the automatic acquisition of a certain number of points with the touch probe for every surface to be studied; the number of detected points indicated below follows common practices and rules of thumb.
• Vertical surfaces are usually sufficiently described by 10 points each (surfaces S1 to S8). • Inclined surfaces are usually sufficiently described by 15 points each (surfaces I1 to I8). • Cylindrical features are sufficiently described by a minimum number of 12 points each to a maximum number of 36 points each, depending on their orientation and dimensions: hence, cylinders C4 and C5 were measured by acquiring 12 points each, cylinders C2 and C3 by acquiring 20 points each, and C1 by acquiring 36 points. • Horizontal planes are commonly established by acquiring 10 points each (planes P1 to P3).
The measurement process took place after the complete setting of this automatic procedure on the aforementioned software of the CMM. The time interval required for a single whole procedure was approximately 10 min per part. The geometric deviations measured for the several part features are reported in Table 6.

Statistical analysis of geometric deviations data
Consistently with Table 2, the extracted data are in terms of geometric deviations-hence, the larger the values, the worse. In the same manner, the effect of the analyzed process parameters will be considered as positive if it concerns a reduction of the geometric deviation, while as negative if involving an increase of the latter. In relation to the geometric deviations reported in Table 6, and with the aim to investigate how these are influenced by the selected process parameters, the experimental results have been statistically analyzed through ANOVA techniques. In particular, still with respect to Table 6, the variables are as follows: • FLAT P , FLAT S , and FLAT I are related to the flatness of horizontal surfaces P, vertical surfaces S, and inclined surfaces I respectively; • CYL C represents the cylindricity; • CONC CxCy indicates the concentricity of the circular feature x with respect to y; • PAR PxPy and PAR SxSy describe the parallelism among horizontal surfaces Px and Py, and vertical surfaces Sx and Sy respectively; • PERP SxP1 indicates the perpendicularity of the vertical surface Sx with respect to the horizontal surface P1.
All the measurements of the geometric deviations are reported in the tables in Appendix Tables 21, 22, 23, 24, 25, 26, 27, and 28.

Flatness of horizontal surfaces
As visible in Fig. 1 and understandable from Table 6, each test, corresponding to a determined printed part, is characterized by having more than a single feature correlated to the analyzed geometric deviation. By considering, for example, the flatness of the horizontal surfaces, each specimen includes three related features, namely FLAT P1 , FLAT P2 , and FLAT P3 . The objective of this work is to evaluate the influence of the process parameters variation on a determined geometric deviation (e.g., FLAT P ), but independently from the printed feature (e.g., without considering if the affected flatness is the one related to surface P1, or P2, or P3). In this regard, if the measured geometric deviation is not affected by the particular feature itself (e.g., by its position, or its orientation with respect to the print platform, or its extension), every measure of a particular geometric feature can be considered as a test repetition. As regards again the flatness of horizontal surfaces, for example, it is possible to assess that, 27 tests, by varying three process parameters on three levels, have been performed with three repetitions (P1, P2, and P3), for a total of 81 acquired measurements, as in Appendix Table 21.
In order to analyze of the effects of the process parameters on the geometric deviations rigorously, the normal distribution of the acquired data must be satisfied. For verifying the normality assumption, the residuals of the examined data are plotted on a probability plot in which the x-axis represents the residuals, and the y-axis expresses their normal probability percentage. Figure 5 depicts the probability plot, resulting from the normality test, for the measured FLAT P . The red dots symbolize the acquired data; the central line corresponds to the cumulative probability; the two external curves represent the 95% confidence interval (CI) boundaries, which is a typical value employed in the practice [70].
If the normality assumption is verified, and consequently the statistical analysis of the input parameters influence has significance, the results in the probability plot are close to the central line and restricted between the CI curves. By this consideration, Fig. 5 shows a lack of normality of the reported FLAT P data. As described in [71], a lack of normality in the data can be generally due to two different conditions: an effective absence of normality of the data, or to the presence of outliers. The latter are mostly caused by calculation or data coding mistakes and are characterized by a high value of the standardized residuals. The distinction between these conditions is difficult, since it relates to several factors, such as the measurer's experience, and the randomization and sequence of the measurements [71]. In any case, it is possible to remove the data having standardized residuals bigger than 3 or 4 times the standard deviation, and checking again the normality of the remaining set of data, without compromising the reliability of the ANOVA results [71].   Tables 21 to 28 Flatness FLAT P FLAT S FLAT I P1-P3 S1-S8 I1-I8 Table 21  Table 22  Table 23 Cylindricity    A preliminary ANOVA of the data related to flatness of horizontal surfaces highlighted the presence of 9 outliers with standardized residual values, such that they can be eliminated. More in detail, the identified outliers are relative to tests 16 and 17 for the surface P1, and tests 10,12,15,19,20,22,26 for the surface P3. The distribution of the outliers does not underline any correlation between them and the process parameters' values. It can be observed, instead, that most of the outliers are related to the acquired measurements on surface P3, and this can be due to the increased difficulty in the CMM's measuring of such a deep and internal feature.
The normality test of the remaining 81 − 9 = 72 measurements led to the probability plot of Fig. 6. The normal distribution of the data in the absence of outliers is clearly identifiable; hence, performing the ANOVA on this set of data will provide reliable information about the influence of the process parameters on FLAT P .
The results of the ANOVA for FLAT P are reported in Table 7. The first column, named "Source," shows the considered process parameters, i.e., LT, PS, and NC. It is possible to notice that, in addition to the influence of the single process parameter on the response (in this case The degrees of freedom, the sum of squares, the adjusted sum of squares, and the adjusted mean squares of the correspondent source are described by DOF, Seq SS, Adj SS, and Adj MS columns, respectively. The results in the F column are calculated by dividing the adjusted mean squares of the source by the adjusted mean square of the error and are of fundamental importance for verifying or rejecting the test statistics hypothesis that the source significantly affects the response (FLAT P ). When F is higher than the related percentile of F-distribution, evaluable by the DOF of the source and the total number of DOF, the considered source affects the response while, on the contrary, if F is lower than the F-distribution percentile, the source influence is negligible. The last column of the ANOVA table presents the value of the parameter p, which is a more convenient indicator for the assessment of the source influence. The value of p is automatically calculated by the statistic software employed for the analysis as a function of F, the related F-distribution percentile, DOF of the source, and DOF of the Error. Depending on the selected CI, 95% in the present work, when p results to be higher than 1 -CI, meaning 0.05, the assumption of the null hypothesis H 0 is correct, indicating in practice that the source has not significant influence  Table 7, it can be noticed that the only parameter affecting the flatness of the printed horizontal surfaces is the LT (p = 0.033). More in details, by observing the main effects plot of Fig. 7, the flatness tends to grow as LT increases, even by considering the presence of a maximum of FLAT P for intermediate values of LT, leading to a worsening of the final geometric accuracy of the component. The near constancy of PS effect plot and the slope of NC lower than LT underline again the lack of influence of NC and PS on horizontal surface flatness, even though a general increase of FLAT P , when increasing both NC and PS, is detectable.
The analysis procedure has been repeated consistently for the other dependent variables.

Flatness of vertical surfaces
The results of the measurements of the flatness of vertical surfaces FLAT S are reported in Appendix Table 22. The normality test of the related data is shown in the probability plot of Fig. 8, where the verification of the normality assumption is clearly visible. All the acquired data are correctly bounded by the two CI curves.
The analysis of variance for FLAT S (Table 8) underlines that the LT and the NC affect the flatness of the printed vertical surfaces, while PS and all the interactions between the process parameters do not affect it. The examination of the main effects plot (Fig. 9), where the contribution of LT presents a positive slope, the evolution of NC shows a negative slope, and the graph of PS is practically horizontal, confirms the previous assessment. In particular, an increase of the LT worsens the vertical flatness, as expected by the previously  reported bibliographic analysis (Table 1), in which the increase of LT reduces the accuracy of the deposited layer, which consequently negatively affects flatness as well. The increment of NC, instead, yields the structure of the part more robust, enhancing the stability of the deposition of the successive layers, positively influencing FLAT S .

Flatness of inclined surfaces
Appendix Table 23 shows the measurements of flatness of inclined surfaces FLAT I . The normality test has been performed, revealing a lack of normality in the data. Consequently, an ANOVA has been performed permitting to identify and eliminate possible outliers. The probability plot of the set of data without the found outliers is depicted in Fig. 10 and exhibits that the normality assumption is verified. The results of the ANOVA (Table 9) indicate that the single effects of all the three parameters influence the response FLAT I . The interaction between the LT and NC affects it too. The main impacts on the response are those related to LT and NC, as suggested by the lower values of the parameters p, while PS and the LT interaction with NC have a limited effect. This is inferable from the higher slopes of LT and NC in the main effects plot of Fig. 11 as well. Increasing both LT and PS results in an increase of FLAT I due to, also in this case, the reduction of the precision of the deposition technique underlined by other studies reported in Table 1. Once again, incrementing the NC improves the stability of the FDM process, giving rise to a reduction of FLAT I .
The results show that, on average, the FLAT I is more than 15% larger than FLAT S and FLAT P , which, in turn, have comparable values. This result highlights the usefulness of having devised a geometry to measure surfaces that are oriented differently. In addition, this difference can be explained through the staircase effect, which is more pronounced in inclined surfaces. Indeed, in horizontal surfaces, the staircase effect is practically absent. In vertical surfaces,

Cylindricity
The measurements of cylindricity deviations of all the features, vertical and inclined ones, are summarized in Appendix Table 24. Figure 12 and Table 10 show the results of the related normality test and of the preliminary ANOVA, respectively.
The lack of normality and the total independence of CYL C from the process parameters are clearly understandable. In this case, since the standardized residual values of the data external to the CI boundaries are lower than 3 times the standard deviation, they cannot be considered as outliers and cannot be removed from the analysis. Moreover, the fact that CYL C is completely independent from the sources is suspicious. In order to better investigate this situation, a scatter plot of cylindricities data, a graphical representation of them by groups, in which each group represents a determined cylindricity feature, has been obtained (Fig. 13). From the distribution of the data in the scatter plot, a good distinction between the vertical cylindricities, namely CYL C1 , CYL C2 , and CYL C3 , and the inclined ones, CYL C4 , and CYL C5 , is detectable. Due to this, it is not possible to consider the analysis as 5 repetitions of the 27 tests, but it has to be subdivided into two different analyses: the first one   (Table 11) and the main effects plot of Fig. 15 indicate that this geometric deviation is negatively affected by PS while positively influenced by NC. The reason of these behaviors is still ascribable to the reduction of precision due to an increased deposition speed, and to the enhancement of stability with a higher NC. Figure 16 illustrates that the normality assumption for the data of the cylindricity of the inclined holes CYL Ci is confirmed.
The ANOVA of CYL Ci demonstrates that, as for CYL Cv , the cylindricity of the inclined holes are significantly influenced by PS and NC (Table 12).
Refining the analysis by the evaluation of the main effects plot for CYL Ci of Fig. 17, the negative effect of a PS increase can be assessed. The p value of the PS influence in Table 12 is lower than the one calculated in Table 11. Thus, the effect of PS is more prominent on the cylindricity of the inclined holes with respect to the vertical ones. This can be explained by considering that, in the case of inclined holes, a cantilever effect takes place between two subsequent deposited layers. An increase of the PS led to a shorter cooling time of the layer that, being subjected to its own weight, and having a lower mechanical resistance, results to be less stable, decreasing the feature precision. Moreover, even if the LT does not significantly affect CYL Ci , the cantilever effect is increased when LT is greater, as visible by the positive slope of LT curve in Fig. 17. The contribution of NC to CYL Ci shows to have an optimum condition for the intermediate value of 2, while the highest value worsens the cylindricity quality. This behavior cannot be explained at present and should be further investigated.

Concentricity
The data related to the measurements of concentricity between the different cylindrical features (Appendix Table 25) complies with the normality assumption (Fig. 18). CONC-CxCy is greatly influenced by LT and PS, as highlighted by the ANOVA analysis of Table 13. The positive slopes of the curves in the main effects plot (Fig. 19) reveal that both the influencing process parameters have negative effects on the  Table 14 report the probability plot, the main effects plot, and the ANOVA results for the parallelism between the horizontal surfaces PAR PxPy (Appendix Table 26), respectively. Since the normality assumption of the data is verified, the ANOVA results, revealing a significant influence of LT and NC on PAR PxPy , can be considered reliable. Figure 21 indicates that increasing the influencing parameters, the parallelism of horizontal surfaces rises as well. This behavior has been already observed for FLAT P and is due to the correlation between the two horizontal features. The medians of PAR PxPy as a function of LT and NC show a maximum for the intermediate values of these latter (LT = 0.20 mm, NC = 2), which indicate a non-monotone evolution. In these positions, anyway, the dispersion of the acquired data is high, as visible by the large measure intervals; thus, the true median could result lower than represented, leading to a monotone function of parallelism, and indicating a general worsening of the parallelism when LT and NC increase.

Surface parallelism for vertical surfaces
The probability plot of the data (Appendix Table 27) of parallelism between vertical surfaces (Fig. 22) exhibits normality.
Looking at the related ANOVA results (Table 15), all the effects of the single process parameters are significant, while their interactions do not affect PAR SxSy .
Among the process parameters, the most influencing one is NC, as suggested by the lowest value of p and the highest slope of the main effect plot curve (Fig. 23). Because of the deposition precision reduction when increasing both LT and PS, these latter negatively affect the vertical surface parallelism, while the increased robustness induced by an augmented NC has a positive effect on it.

Surface perpendicularity
The normality test for the perpendicularity of vertical surfaces S with respect to the horizontal surface P1 has been passed, as reported in the plot of Fig. 24. The related experimental measurements are reported in Appendix Table 28.
From the ANOVA results (Table 16), the dependency of PERP SxP1 from all the single effects of the process parameters is detectable. Also in this case, LT and PS negatively affect the response, while this is positively influenced by the increased stability deriving from a high NC (Fig. 25).

Regression analysis
Starting from the ANOVAs of the considered responses, the process parameters mainly affecting them have been identified. With the intent of developing a set of mathematical models to forecast the geometric deviations as a function of the affecting parameters, a regression analysis for each response has been performed. This led to the Eqs. from (1) to (9): For assessing the capability of the introduced mathematical models to correctly estimate the related geometric deviations, they were applied for calculating these latter by considering the combination of LT, PS, and NC and comparing the model outcomes with the experimental ones. Table 17 reports the minimum (e min ), maximum (e MAX ), and percentage (e % ) errors resulting from this comparison. For each geometric feature and process parameters' combination, the difference between measured and modelled data was calculated. Among these, e min and e MAX represent the minimum and the maximum values achieved, respectively. The average of the single percentage errors deriving from the calculated differences was defined as e % .
Despite the presence of a low e % for geometric features such as FLAT I , CYL Cv , and CONC CxCy , the calculation errors for FLAT S and PAR PxPy are above 20%, resulting in an average percentage error of 15%. Even if greater errors in FLAT S and PAR PxPy estimation can be made with respect to the evaluation of the other geometric deviations, this average value is still good, indicating a general good prediction ability of the mathematical models. In this manner, an indication of the final part geometric characteristics, once the process parameters have been selected, can be achieved. The process parameters optimization can be accomplished as a function of the desired geometric characteristics. The occurrence of high e % values for some responses is ascribable to the fact that, as observed in Table 1, several process parameters affect the printed part quality. In addition to LT, PS, and NC, other parameters, such as raster-to-air gap, infill density, nozzle temperature, have a significant influence on the geometric deviations. Since the proposed models concern only the variation of the process parameters reported in Table 4, it is not possible to consider the effects of other parameters. Hence, to improve the models' performances, further experiments foreseeing different printing parameters should be performed.

Printing times
The PTs, expressed in minutes for all the performed tests, are reported in Table 5. The values satisfy the normality assumption (Fig. 26). As expected, the ANOVA (Table 18) indicates that all the process parameters significantly affect the production time. Particularly, when increasing the LT and the PS, the time needed for the realization of the part is reduced, while if the NC is increased, the FDM process requires more time (Fig. 27).
The regression analysis of the data permitted to derive the regression equation able to calculate the PT as a function of the process parameters (Eq. (10)).
The comparison of the PTs calculated by Eq. (10) and the experimental ones (Table 19), gives a mean of the  Tables 17 and 19, which summarize the calculation errors by applying the derived regression models in Eqs. (1-10), confirm their reliability in the estimation of the geometric deviations and the PTs as a function of the employed range of the process parameters. Therefore, the proposed models can be further analyzed for optimization scopes in compliance with the objectives stated in Sect. 1. Figure 28 shows the evolution of the estimated geometric deviations as a function of the PT. As expected, the tendency of a general enhancement of the quality, meaning a reduction of the geometric deviations, when the PT increases, is clearly visible. As previously observed, this is due to the lowering of LT and PS, and to the increase of NC, that improves the FDM deposition process while at the same time increasing the PT (Fig. 27). Consequently, in order to minimize geometric deviations, a higher PT is unsurprisingly required.

Effects of parameters on quality vs. printing time
To better understand how the process parameters affect the PT and the totality of the geometric deviations, their values have been normalized and represented as a percentage of their amounts. More in detail, the normalization of the employed process parameters has been performed by applying Eq. (11): • PP i is the value of the process parameter to be normalized, • PP low and PP high are the low and high value of the process parameter (Table 4), respectively.
Geometric deviations and time have been normalized using Eq. (12): where • DT % is the normalized geometric deviation or time, • DT max is its maximum overall value, and • DT i is the value to be normalized.
The applied normalization permits to represent all the response parameters, namely, geometric deviations and PT, in the same plot to evaluate the influence of process parameters on them clearly. Figures 29, 30, and 31 report the behavior of the normalized response parameters as a function of normalized LT, PS, and NC, respectively. In these plots, the dotted lines represent the evolution trend of responses, and their agreement with the previously depicted main effects plot is observable. In addition to this, the slope coefficient sl of each trend line is indicated.
The significance of the process parameters' influence can be assessed by the comparison of sl. In particular, the higher the absolute value of sl, the higher the significance in which it is influenced. Moreover, a negative value of sl indicates a geometric deviation or a PT reduction, meaning an improved quality, or a quicker process, respectively, while the opposite applies in case of a sl positive value.
The values of sl are summarized in Table 20, where, for a quicker identification of the most influencing parameters, the background of each cell is colored in a scale from light to dark gray, as a function of the sl absolute value. Through this table, an overall ranking of importance of the effects of process parameters on quality and PT can be assessed.    Overall, the most affecting parameter results to be NC, followed by LT and then PS.
Once the weight of importance of the process parameters on the quality and time responses is identified, in order to select the best combination of the values of the process parameters, an optimization process was performed. The target of the optimization was to maximize the accuracy of the product considering the possibility of reducing the production time as well. Under these assumptions, the optimization setup concerned the minimization of all the geometric deviations and PT. The results of the optimization phase are visible in Fig. 32. In the first four rows, the high and low values of process parameters are reported in black, while their optimized values are highlighted in red under the indication "Cur." The first column of the subsequent rows shows the examined response; its estimated value, as a function of the regression models of Eqs. from (1) to (10), in the correspondence of the optimized process, in blue; and its desirability. The latter marks the closeness of the estimated value of the response with respect to the desired one. In practice, since the objective of the optimization is the response minimization, as indicated by the term "Minimum" in Fig. 32, this means that a desirability value equal to 1 represents a minimized response, while the response gets worse if the value becomes lower. In the right hand part of Fig. 32, the evolution trends of the responses, while the process parameters increase, are depicted with black lines. Also in this case, the agreement of these trends with the ones of Figs The optimized values of the process parameters are LT equal to 0.16 mm, the lowest LT value, PS equal to 89.97 mm/s, close to the mid-range value of PS, and NC equal to 2.92, close to the highest NC value. These results underline again the main importance of the influence of LT and NC as asserted in Table 20. The optimized PT results to be equal to 180 min, which is not properly minimized as it corresponds to the 80% of the maximum PT. This is correlated to the fact that, during the optimization phase, higher importance has been given to the final quality of the printed component. As previously observed in Fig. 28, the quality improves when the PT increases. A minimization of the PT surely reduces the AM machine hourly costs but, at the same time, leads to the worsening of printed part quality, which implies mandatory costly and time-consuming post-process operations potentially overcoming the previously saved expenses. In this perspective, a PT of 180 min can be deemed acceptable as it provides for the highest achievable quality in the explored range of process parameters.

Conclusions
It is acknowledged that AM allows the extension of the design space, especially in terms of shape freedom. However, the quality and performances of AM-ed parts are simultaneously affected by both the design itself and the chosen process parameters for its concrete realization. Optimizing the interdependent design and fabrication is also a prerogative of the FDM technology [72], which is both established for RP scopes and increasingly used for the manufacturing of end products. In this context, the literature has so far failed to address the effect of process parameters on geometric accuracy, despite GD&T is considered among the relevant fields for the standardization of AM processes [68]. Contextually, the achievement of quality characteristics is commonly counterbalanced by long 3D printing times (the results shown in Sect. 4 are of no exception), which should be also considered in optimization tasks.
The present paper represents an original attempt to widely address the role played by FDM process parameters in geometric accuracy, while contextually considering the process duration. To this scope, the design of experiments has included a full factorial combination of three considered process parameters manipulated in three levels. Five typologies of geometric deviations and the printing time have been used as response variables. For all these typologies, the paper reports results, statistical analysis, and the authors' comments (Sect. 4) whenever they believed that the effects of process parameters on accuracy could be explained plausibly. Evidently, the statistical analyses provided significant correlations, but those cannot be interpreted as cause-effect relationships. This implies that the authors' interpretation can be challenged by other scholars or experts. Relevant similarities and differences with past studies have been highlighted as well in Sect. 4. The results include a proposal to optimize process parameters to target accuracy and time, although the former was prioritized. From a methodological viewpoint, this optimization is a plain attempt to balance quality and time aspects in a FDM process, to be possibly replicated with different quality factors, priorities, and, possibly, mechanical performances.
Within the results, the potential effect of the chosen process parameters on geometric accuracy hypothesized by the authors (see Sect. 4.4) has been confirmed. The NC, which has been poorly investigated in previous studies concerning the quality of FDM 3D-printed parts (see Table 1), turned to be the most impacting factor overall. Nevertheless, it has to be noted that the effect of the process parameters is uneven if the different kinds of geometric deviations are considered. This calls into the question the need to consider each situation separately based on the design of the part and the critical geometric tolerancesthe 3D-printed part used here was designed for experimental purposes only.
In addition to this, the outcomes highlight that the position of surfaces is a further factor to be considered, e.g., the investigated process parameters affected differently the flatness of horizontal and vertical surfaces. The numerous relations that have been studied offer abundant insights, but the data made available lends itself to even more detailed analyses, e.g., differences of effects across planar horizontal surfaces according to their distance from the printing plane.
The research is nevertheless subjected to several limitations, which are considered unavoidable since the topic has been practically neglected hitherto. Markedly, a number of factors should be considered that could invalidate the presented results if a similar study is replicated. These factors include a number of aspects or parameters chosen in a partially arbitrary way or for convenience; examples follow: • The geometry and size of the part; here, it can be however hypothesized that the topology would affect process times more than the effects of process parameters on geometric accuracy. • The material used (PLA). • The 3D printing FDM device.
As the number of possibly manipulated process parameters is plainly larger than the set of the chosen ones (see Table 1), the most critical parameter for geometric accuracy cannot be properly identified. Future studies should aim to contextually investigate different parameters and validate the effects of the already studied ones. In this respect, the authors are available to share additional materials, e.g., the CAD model of the used part, to allow the replication of the experiment by other research groups. Authors' future work includes akin studies focusing on materials used in FDM and characterized by better mechanical performances, e.g., fiber-reinforced filaments.