Prediction of the mechanical properties of 3D-printed parts using the Gaussian Regression Process (GRP) Machine learning model.

Akessa, Adugna D1 Tucho Wakshum M2 and Ren Chao2

1,2 Stavanger University, Stavanger, Norway

adugna.d.akessa@uis.no

wakshum.m.tuch@uis.no

chao.ren@uis.no

Abstract. Multiple parameters, such as raster angles, printing temperature, printing speed, infill density, and infill pattern, significantly affect the mechanical properties of 3D-printed parts. Identifying the optimal combination of parameters from a multitude of options is a difficult task. In response to this issue, a Gaussian Process Regression (GPR) machine learning model was developed to predict the flexural strength and ductility of Polylactic Acid (PLA+) parts produced through Fused Deposition Modeling (FDM). Printing speed, bed temperature, and raster angle were selected as input variables, and a limited dataset (13 parameter combinations) was used to train the GPR model. The model effectively predicted these properties and identified the top five parameter combinations to maximize flexural strength and ductility among the thousand combinations. Test specimens were fabricated using the predicted parameters and experimentally evaluated to validate the results. The study revealed that experimental values deviated by less than 2% and 0.3% for flexural stress and strain, respectively, from the predicted results, with experimental results slightly lower. This confirms that the GPR model is highly effective in accurately predicting the mechanical properties of additively manufactured components. The findings present a reliable approach for optimizing FDM process parameters, minimizing the need for extensive experimental trials, and accelerating the design and manufacturing process.

Keywords: Machine Learning, FDM Modeling, Process optimization

1 Introduction

Additive manufacturing (AM) is a method in which a 3D model from a computer-aided design (CAD) is fabricated in a layer-by-layer mode [1]. Three-dimensional (3D) printing technology has gained significant prominence recently and introduced a new flourishing innovation in the design and fabrication of components. Complex geometry, which conventional manufacturing methods cannot produce, can be manufactured using AM with low wastage and quickly. Parts manufactured by additive manufacturing processes are used in biomedical fields for tissue and organ replacement, in the automotive industry for lightweight structures, in construction for load-bearing applications, and for prototyping in the educational field [2-5].

The 3DP technology is classified as solid-based, powder-based, or liquid-based depending on the form of the raw materials used. Laminated object manufacturing (LOM) and fused deposition modeling (FDM) are among the solid-based 3DP technologies, whereas powder-based technologies include selective laser melting (SLM), selective laser sintering (SLS), and adhesive binder jetting. The stereolithography apparatus (SLA) is a liquid-based technology [6].

In contrast to other 3D printing techniques, fused deposition modeling (FDM) is the most popular and widely used method due to its simplicity of operation, ease of accessibility, improved accuracy, low energy consumption, and cost-effectiveness compared to other methods. It operates by feeding a thermoplastic filament through a heater element, where the filament is melted, pushed through a nozzle, and extruded in layers for deposition until the model is entirely built [7, 8].

FDM produces parts from thermoplastic materials that are closer to the design shape, with over 90% material utilization compared to traditional manufacturing processes. Thus, FDM can be an environmentally friendly process, a just-in-time manufacturing process, and can reduce the need for transportation, with on-site production [9]. FDM performance can be enhanced through improving the poor mechanical properties and overcoming the difficulty of selecting process parameters that enhance the engineering application of FDM-produced parts. This can be achieved by studying input parameters such as printing speed, orientation, layer thickness, nozzle and printing bed temperatures, infill percentage, pattern, and other parameters [10-12].

Several studies have been conducted to optimize the mechanical properties of components produced by FDM technology by improving different printing process parameters using various approaches. Machine learning (ML) is a timely solution for enhancing the mechanical properties of parts produced by the FDM manufacturing method. It can reduce experimental time and is a vast domain where data is mined, allowing a machine learning model to learn independently [13].

The ML model can be trained to predict output data based on the required input data, with the capacity to handle vast amounts of data. It can predict uncertainty and solve the complex relationship between the input variable (3D printing process parameter) and the output variable (product performance) [14]. In 3D printing, ML is widely used to predict optimal process parameters for better mechanical properties, such as flexural strength, tensile strength, hardness, etc. Flexural strength refers to the resistance of materials to transverse forces. Bending strength is applied to brittle materials like plastics, composites, ceramics, etc. [15].

This study investigated the influence of various process parameters on the flexural strength and strain, utilizing machine learning (ML) to predict the optimal combination of process parameters, thereby achieving improved mechanical properties, specifically enhanced bending strength.

Agarwal et al (16) developed a machine learning model to predict the mechanical strength of orthopaedical bone screw manufactured by FDM 3D printing method using different printing parameters (i.e., infill pattern, infill density, wall thickness and layer height) so that it can have better performance in load bearing capacity [16]. Gunay (17) modeled the tensile and bending strength of Polylactic Acid (PLA+) parts produced by the FDM 3D printing method using a regression model where raster angle, infill density, and printing speed as input variables, and concluded that increasing printing speed decreases tensile and bending strength as well higher raster angle decrease the tensile and bending strength [17]. Jalil et al. (18) studied different ML models to overcome the mix design challenge of 3D-printed concrete parts by optimizing the bending and tensile strength. They studied Gaussian Process Regression (GPR), Decision Tree Regression, Support Vector Machine learning, and XGBoost Regression models to evaluate the performance of each model that can optimize better based on six different mix proportions [18].

Veeman et al. (14) developed different machine learning (ML) models to predict the hardness of additively manufactured acrylonitrile butadiene styrene (ABS) based on infill density, layer thickness, print orientation, and raster angle [15]. Trivedi et al. (19) developed an ML model for predicting the tensile strength of 3D-printed parts, concluding that higher tensile strength is obtained at lower orientation, higher extrusion temperatures, and higher layer thickness [19].

This study investigates the influence of process parameters on the flexural strength and strain properties of 3D-printed PLA+ parts. Sixty-five experiments were conducted to determine the impact of the following influential printing parameters: printing speed (PS), printing plate temperature (PT), and raster angle (RA). The experiment design was generated using the Minitab software, and three levels were used for each parameter. All experimental datasets were used to train the model to predict the output results. The ML model used was a Gaussian Process Regression (GPR) due to the limited size of the dataset and the non-linearity of the relation between input and output variables. From the training, we predicted the optimal combination of input variables that helps optimize the output variables and evaluated the accuracy of the ML model using experimental work.

2 Materials and methods

2.1 Materials used

In this study, specimens were printed using PLA+ (white), a material manufactured by Elefun Norway. PLA, a biodegradable aliphatic polyester sourced from lactic acid, finds applications in various fields, including biomedical, packaging, textile fibers, and technical items [20]. The properties of the material provided by the manufacturer are listed in Table 1.

Table 1. Properties of PLA+ material used in the study.

Characteristics	Values
Extrusion temperature (°c)	205 -230
Softening temperature (°c)	50
Density (g/cm³)	1,23 – 1,25
Flexural strength (MPa)	55,3
Moisture absorption (%)	0,5 - 50
Filament diameter (mm)	1,75
Elongation at failure (%)	3,8
Printing accuracy (%)	±0,1
Tensile strength at break (MPa)	57.8

2.2 The specimens

The specimens studied were 3D printed according to the ASTM D 790 standard [21] using a Bambu Lab X1 FDM 3D printing machine. The model was created using Autodesk Inventor 2025. The open-source Bambu Studio slicer controls printing parameters and exports the G-code of the generated 3D model to the printer [see Fig.1].

Fig. 1. The geometry and dimensions of the flexural test specimen.

The Bambu Lab X1 FDM 3D printing machine offers a building volume of 256 x 256 x 256 mm³, achieves a maximum printing speed of 500 mm/s, and features a nozzle diameter that varies from 0.2 to 0.8 mm [22]. To reduce the likelihood of errors, five test specimens were printed for each test, and the printing parameters examined in this study are detailed in Table 2.

Table 2. The variable parameters considered in the study, along with their corresponding value ranges.

Process parameters	Range of values
Printing Speed (mm/s)	300, 400, 500
Printing plate temperature (°c)	60, 70, 80
Orientation angle (°)	0, 45, 90
Infill density (%)	100
Infill pattern	Monotonic

2.3 Mechanical testing

The 3D-printed specimens were subjected to a 3-point bending test according to ASTM D 790 standard (22). The specimen was positioned on two rollers and was loaded at the midpoint of the loading noses in the column. The testing was conducted using an Instron 5985 universal testing machine, manufactured in the USA, equipped with a 10 kN load cell at room temperature and pressure, at a strain rate of 1 mm/min.

2.4 Experimental design

The effects of 3D printing process variables, such as PS, PT, and RA of the filaments, were considered in the study. Each variable was analyzed at three different levels, and the printed conditions were generated randomly using Minitab software, as shown in Table 3, which serves as a test dataset. The input parameters for all tests included printing platform temperature (ranging from 60 to 80 °C), printing speed (ranging from 300 to 500 mm/s), and orientation angle (ranging from 0 to 90 °). Based on these input variables, 13 potential combinations were generated, and 5 test pieces were printed for each combination to minimize the likelihood of error. The algorithm for the study was shown. (see Fig. 2)

Fig. 2. The Algorithm for the proposed work.

The experimental tests were carried out at the Mechanical Test Laboratory of the University of Stavanger to determine the flexural strength and strain values.

Table 3. 3D printing parameter setting.

No.	PS (mm/s)	PT (°c)	RA (°)	Infill density (%)	Nozzle Temperature (°c)	Infill pattern
1	300	60	45	100	220	Monotonic
2	300	70	0	100	220	Monotonic
3	300	70	90	100	220	Monotonic
4	300	80	45	100	220	Monotonic
5	400	70	45	100	220	Monotonic
6	400	80	90	100	220	Monotonic
7	400	80	0	100	220	Monotonic
8	400	60	0	100	220	Monotonic
9	400	60	90	100	220	Monotonic
10	500	70	0	100	220	Monotonic
11	500	80	45	100	220	Monotonic
12	500	60	45	100	220	Monotonic
13	500	70	90	100	220	Monotonic

2.5 Machine Learning

Various machine learning models are employed depending on the quantities and types of dataset samples available. For example, artificial neural networks (ANNs) utilize multiple hidden layers and nonlinear functions to model the independent and dependent variables; however, they require large datasets. One-dimensional convolutional neural networks (CNNs) can be used in regression tasks when the dataset is time-series or sequential and contains many samples [23]. Given the limited dataset available, a non-linear relationship between input and output variables, and the simplicity of the machine learning tools, we utilized the Gaussian process regression model to analyze and predict the relationship between the dependent and independent variables. There are still machine learning models (e.g., Support Vector Regression, Decision Tree Regression (DTR), and XGBoost) that fulfill these requirements; however, Gaussian process regression (GPR) is the best fit for this study because of its ability to quantify uncertainty in predictions, representation, and flexibility [24].

This research aimed to investigate the influence of printing parameters on 3D printed	PLA parts using a GPR machine learning model and to predict the optimal parameters, thereby achieving maximum flexural strength and strain. GPR is a probabilistic supervised machine learning model used in regression analysis and classification, which makes predictions based on prior knowledge (Kernels) and provides an uncertainty measure over its predictions. It is widely used due to its ability to quantify uncertainty in predictions, representation, and flexibility [25]

In modeling practice, no specific functional form is considered. Nevertheless, it models the dataset's distribution directly using Bayesian inference to make predictions based on the observed data, as shown in Equation (1) [25]. A variable X is Gaussian or normally distributed with mean µ and variance and its probability density function is:

Px(x) = exp	(1)

Where X is a random variable and x is a value.

GPR assumes that the output variable y is a function of the input variable x, in that given a training data set D = {(xi, yi) | i = 1..., n}, which can be modeled as a Gaussian process. It is specified by a mean function (x) and a covariance function K (x, x’), which are used to estimate the conditional probability distribution of the output variable y [26]. The design matrix X is used to define the input space of the Gaussian Process, and the vector of desired output y is used to train the model. The primary assumption of GPR is that the output y is computed as y = f (x) + Where f(x) is the unknown proper function and is the additive Gaussian noise with mean zero and variance It also assumes that x follows a Gaussian Process, and therefore, the predicted output y* for a new input x* is also a Gaussian distribution [26].

A total of 13 datasets were utilized, which was a limited number; consequently, a GPR machine learning model was implemented. The independent variables were employed to forecast the dependent variables and to determine the optimal combination of five input variables that resulted in maximum flexural strength and strain. The relationship between data points is characterized by the radial basis function/kernel. Test pieces for the five maximum output variable combinations were printed and tested experimentally based on the predicted input variable combinations, and the results from the test were compared with the predicted output variables to evaluate the model's accuracy.

3 Results and discussions.

The experimental test results are presented in Table 4, and this study investigated the correlation between the input and experimental result or output variables using free Colab Python software. Fig. 3 shows a heatmap generated using an algorithm from the Seaborn library.

Table 4. Experimental test results.

Number	PS (mm/s)	PT (°c)	RA (°)	Stress (MPa)	St. deviation (%)	Strain (%)	St. deviation (%)
1	300	60	45	67,32	1.62	3,84	0,03
2	300	70	0	72,44	0,65	3,85	0,03
3	300	70	90	62,12	1,72	3,51	0,29
4	300	80	45	80,89	5,2	3,80	0,06
5	400	70	45	67.15	0,88	3,74	0,23
6	400	80	90	61,44	2,04	3,30	0,23
7	400	80	0	74,56	1,58	3,70	0,16
8	400	60	0	79,05	0,55	3,45	0,03
9	400	60	90	61,86	3,69	3,20	0,23
10	500	70	0	70,93	1,69	3,67	0,01
11	500	80	45	67,55	0,33	3,63	0,01
12	500	60	45	66,74	0,17	3,64	0,01
13	500	70	90	64,64	1,15	3,40	0,01

The temperature of the 3D printing plate is positively correlated with both Flexural stress and flexural strain, with flexural stress demonstrating a stronger correlation than flexural strain. Conversely, both printing speed and raster angle exhibit a negative correlation with flexural stress and strain. There is a strong negative correlation with the raster angle and a weak negative correlation with printing speed. This suggests that the lower raster angle and printing speeds lead to improved flexural stress and strain measurements.

Fig. 3. Heatmap showing the correlation between input and output variables of datasets.

This indicates that the specified parameters of the printing process significantly influence the flexural stress and strain of components manufactured from PLA using FDM methods. The findings are consistent with a previous study [26], which suggests that an increase in printing speed leads to a decrease in flexural stress and strain. Additionally, a higher raster angle corresponds to a reduction in flexural stress and strain. However, increasing the temperature of the printing plate results in an increase in flexural stress and strain.	It was noted by (27) that a higher raster angle orientation reduces the flexural stress, whereas a lower raster angle increases its flexural stress [27].

Data normalization was performed to enhance the model's accuracy. This process involved scaling the numerical features of the dataset through min-max scaling as outlined in Equation (2) [27].

(2)

Where is the normalized value of the parameter, x is the original value, x(min) is the lowest value of the parameter, and x(max) is the highest value of the parameter.

The accuracy of the GPR model's prediction was evaluated by dividing the dataset into training and testing sets. The training set was used to develop the predictive model, while the test set was used for its validation. This study assessed prediction accuracy through the mean square error (MSE) and R-squared (R2) metrics.

MSE is the most used method, and the model evaluation is done by measuring the average square magnitude of errors generated by the models according to Equation (3) [30]. This approach squares the deviations leading to the amplification of the error, thereby confirming the prediction model's effectiveness.

(3)

(4)

Where n is the number of datasets, ypre is the predicted value, and yactual is the actual value. The higher the values of MSE are, the less accurate the model is however, with MRE of 0.6%, the model demonstrates better accuracy. Furthermore, the R-squared value defined in Equation (4) [27] for evaluating the regression model ranges from 0 to 1, with higher values indicating greater accuracy.

Table 5. Experimental data and GPR predicted data.

Run	Input variables	Experimental Results	Predicted results	Absolut Error
PS	PT	RA	Stress (Mpa)	Strain (%)	Stress (Mpa)	Strain (%)	Stress (Mpa)	Strain (%)
1	300	60	45	67.32	3.84	67.12	3.83	0.2	0.01
2	300	70	0	72.44	3.85	72.44	3.84	0	0.01
3	300	70	90	62.61	3.53	62.02	3.52	0.09	0.01
4	300	80	45	80.89	3.8	80.72	3.79	0.17	0.01
5	400	70	45	67.15	3.74	67.72	3.72	0.03	0.02
6	400	80	90	61.44	3.3	61.26	3.29	0.18	0.01
7	400	80	0	74.56	3.7	74.55	3.67	0.01	0.03
8	400	60	0	79.05	33.45	79.03	3.44	0.02	0.01
9	400	60	90	61.86	3.2	61.76	3.19	0.1	0.01
10	500	70	0	71	3.67	71	3.65	0	0.02
11	500	80	45	67.55	3.63	67.53	3.62	0.02	0.01
12	500	60	45	66.74	3.64	66.72	3.64	0.02	0
13	500	70	90	60	3.4	60	3.39	0	0.01

The R-squared value for this model is 99.16%. The predicted and actual values of the dataset are presented in Table 5 [28]. Fig. 5 and Fig. 6 compare plots of the actual and predicted datasets for flexural stress and strain, respectively.

Fig. 4. Comparison between experimental and predicted flexural stress.

Fig. 5. Comparison between experimental and predicted flexural strain.

To identify the best combination of input variables that yield an optimal output variable, the Radial Basis Functions (RBF) kernel was employed. The optimal input parameters (PS, PT, and RA) were rated to find the optimal combination to improve the output parameters (Flexural stress and Strain). Based on the study's objective, the GPR machine learning model underwent multiple training iterations using Colab Python free software, resulting in the identification of the optimal input variable combination. Among these combinations, the top five combinations were considered, along with their corresponding output variables, and an experimental test was conducted to validate the results presented in Table 6.

Table 6. Experimental data and GPR predicted data.

Input Variables	Predicted Results	Experimental Results	Error
PS	PT	RA	Flexural Stress (MPa)	Flexural Strain (%)	Flexural Stress (MPa)	Flexural Strain (%)	Flexural Stress (%)	Flexural Strain (%)
300	80	40	85,2	4	83,2	3,93	2	0,07
300	80	45	85	4,2	80,43	3,9	4,57	0,3
300	78	40	84,7	3,92	79,68	3,55	5,02	0,37
300	78	45	84,4	4,1	79,4	4	5	0,1
300	80	35	84,8	3,6	79,35	3,54	5,45	0,06

Table 6, Fig. 6, and Fig. 7 illustrate the relationship between the top five predicted and experimental results for both flexural stress and strain. The experimental results and the machine learning model show that increasing printing speed (PS) and Raster angle (RA) results in lower flexural stress and strain values. In contrast, printing plate temperature has a positive effect on flexural stress and strain values.

Fig. 1. Plot of the top five predicted results of flexural stress and their experimental results.

Fig. 7. Plot of the top five predicted results of flexural strain and their experimental results.

Upon reviewing the GPR machine learning model, it is observed that the R² and MSE values are 99.12% and 0.6%, respectively. This indicates that the model efficiently predicts the correlation between the input variables (PS, PT, and RA) and the output variables (Flexural Stress and Strain). As shown in Figs. 6 and 7, the predicted results exceed the experimental results by a maximum of about 2% and 0.3% for flexural stress and flexural strain, respectively. However, a comparison of the initial experimental result in Table 5 with those in Table 6 reveals improvements in flexural stress and Flexural strain of 3 MPa and 0.16%, respectively.

This research indicates that the flexural strength and strain measurements improved compared to the figures in the manufacturer's technical data sheet, as illustrated in Table 1. Consequently, flexural strain and strength have increased by 0.2% and 28 MPa, respectively. The findings present a reliable approach for optimizing FDM process parameters, reducing the necessity for extensive experimental trials, and accelerating the design and manufacturing process.

Conclusion

In this work, flexural specimens were produced using the FDM 3D printing method, varying the printing process parameters. The influence of printing parameters on the flexural strength and strain was evaluated. The Flexural strength and strain were predicted using the GPR ML model. The findings present a reliable approach for optimizing FDM process parameters, thereby reducing the need for extensive experimental trials and accelerating the design and manufacturing process.

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