Flexibility analysis and process parameter optimization of TPU materials printed by FDM

ABSTRACT

The manufacturing sector is increasingly focusing on Fused Deposition Modeling (FDM) methods. However, existing parameter optimization for thermoplastic polyurethane (TPU) in FDM lacks systematic screening, quantitative interaction analysis, and rigorous stability verification. To address these deficiencies, this study proposes a systematic optimization method addressing these three core gaps. It investigates the effects of four FDM parameters—printing speed (A), printing temperature (B), infill rate (C), and layer height (D)—on TPU flexibility, which is characterized by elastic modulus , where lower values indicate superior fexibility. The methodologies employed include single-factor ANOVA, Duncan’s test, SPSS-analyzed orthogonal experiments, and verification tests. All parameters significantly affect the elastic modulus (p < 0.01), with the significance order of B > A > A × B (p < 0.05). The optimal parameters identified are 55 mm/s printing speed and 200 ℃ printing temperature, with a coefficient of variation of 0.40% (indicating stable performance). This study provides support for TPU flexibility optimization and FDM forming stability enhancement.

Keywords:
Fused Deposition Modeling (FDM); Thermoplastic Polyurethane (TPU); Optimization of Process Parameters; Elastic Modulus.

1. INTRODUCTION

Fused Deposition Modeling (FDM), a mainstream additive manufacturing technology, has achieved signiffcant breakthroughs in the global manufacturing industry in recent years. By leveraging the unique advantage of layer-by-layer stacking fabrication, FDM enables rapid production of complex structural components without the need for intricate molds. Its application potential has been widely recognized across high-value industries such as aerospace, biomedical, and automotive [¹]. FDM technology not only meets the processing requirements of traditional rigid materials but also continuously expands its adaptability and application scope in soft material manufacturing, driven by the growing industrial demand for customized soft materials. Thermoplastic polyurethane (TPU), a novel soft material, exhibits excellent elasticity, wear resistance, and biocompatibility, making it promising for wide applications in medical devices and high-end component manufacturing [², ³, ⁴]. FDM technology effectively supports the personalized and customized manufacturing of TPU parts, enabling the precise fabrication of medical aids and lightweight high-end components [⁵]. However, the industrialization of FDM-printed TPU is hindered by the bottleneck of inadequate process parameter optimization [⁶]. Key process parameters, including printing temperature, printing speed, and layer height, directly determine the forming quality and mechanical properties of TPU parts. Concurrently, TPU exhibits significant differences in flexibility compared to conventional rigid materials. The existing FDM parameter system optimized for rigid materials is difficult to directly apply to TPU printing, often resulting in poor interlayer bonding and forming defects. These defects significantly degrade the flexibility and service performance of printed parts. Therefore, systematically exploring the optimization method of FDM process parameters for TPU printing holds significant theoretical value and engineering importance for enhancing the performance stability of TPU parts and facilitating their large-scale industrial application.

Current domestic and international research on FDM-printed TPU materials primarily focuses on mechanical property analysis and the exploration of individual parameter influences. Hu et al. [⁷] obtained the zero-energy characteristics of TPU printed models under different process parameter configurations through static experiments, providing foundational data support for parameter optimization. Xie Bowen et al. [⁸] employed a combination of single-factor and orthogonal experiments, identifying material hardness and infill rate as the primary and secondary factors influencing TPU flexibility, thereby clarifying the core direction for flexibility regulation. Lei Jingfa et al. [⁹] noted that FDM process parameters significantly influence the mechanical properties of TPU printed parts, with printing temperature and infill rate as key control parameters. These parameters govern changes in core mechanical indicators such as elastic modulus by affecting interlayer bonding quality, and reasonable regulation of the parameter range can effectively optimize part performance. Chaudhry & Czekanski [¹⁰] indicated that Fused Filament Fabrication (FFF) process parameters exert a significant regulatory effect on the mechanical response of TPU printed parts. Establishing a correlation model between process parameters and mechanical properties enables effective quantification of each parameter’s influence law, providing theoretical and model support for the precise regulation of TPU printed parts’ mechanical properties. Meanwhile, recent studies on additive manufacturing (AM) and Fused Deposition Modeling (FDM) have witnessed remarkable advances in parameter optimization through machine learning (ML) algorithms [¹¹,¹²,¹³,¹⁴]. Relevant research in this area has focused on diverse targets, including energy consumption minimization [¹¹], dimensional accuracy evaluation [¹²,¹³,¹⁴, ¹⁶],FDM raster angle estimation [¹⁵], and performance prediction of fiber-reinforced PLA components [¹⁷]. These studies have achieved high prediction accuracy using various ML models such as Random Forest Regression (RFR), Gaussian Process Regression (GPR), and optimized Artificial Neural Networks (ANNs), demonstrating the effectiveness of ML-based approaches in FDM parameter modeling and prediction [¹¹, ¹³, ¹⁶,¹⁷,¹⁸]. For example, some studies have achieved extremely high accuracy in their specific research fields, with GPR models for energy consumption prediction achieving an R² value of 0.99 [¹¹], and evolutionary algorithm-optimized neural networks for MEX dimensional accuracy prediction resulting in a Mean Absolute Percentage Error (MAPE) of ≤1.3% in the X/Y directions [¹⁶]. In addition, ML has also been successfully applied to predict the mechanical properties of FDM parts and TPU-based components[¹⁷,¹⁸,¹⁹].A systematic comparison of these existing studies (References [¹¹,¹²,¹³,¹⁴,¹⁵,¹⁶,¹⁷,¹⁸,¹⁹]) with the proposed method is presented in Table 1, aiming to clarify differences in optimization methods, core performance metrics, and respective advantages of each research direction.

Table 1
Comparison of existing studies and the proposed method.

RESEARCH TYPE	OPTIMIZATION METHOD/ALGORITHM	CORE PERFORMANCE METRICS	UNIQUE SUPERIORITY
FDM-TPU Flexibility Optimization	ANOVA, Duncan’s test, orthogonal experiment, secondary polynomial modeling	Printing speed: 55 mm/s; Nozzle temperature: 200 °C; Coefficient of variation (CV) = 0.40%; High accuracy in TPU mechanical property prediction	Systematic parameter pre-screening and verification; high relevance to TPU flexibility optimization
Dimensional Accuracy [12,13,14, 16]	[13]: Taguchi method + Artificial Neural Network (ANN); [16]: Neural Network (NN) optimized by evolutionary algorithm; [12, 14]: Additive Manufacturing (AM)/Material Extrusion (MEX) parameter analysis	[13]: ANN R² > 95%; [16]: Mean Absolute Percentage Error (MAPE) ≤ 1.3% (X/Y), 2.3% (Z); [12, 14]: Systematic parameter analysis	High accuracy and robustness; supports AM parameter optimization; useful for FDM regulation
Energy Consumption Prediction [11, 17]	[11]: Machine Learning (ML) models (GPR best); [17]: GPR, XGBoost, ANN, Random Forest Regression (RFR best) + ANOVA	[11]: R² = 0.99; MAE = 0.016; RMSE = 0.022; MSE = 0.00049; EVS = 0.99; [17]: LT = 43.96%, ID = 27.58% as key factors	Extremely high prediction accuracy; identifies key influencing factors
FDM Parameter Prediction [15, 18, 19]	[15]: SVM, GPR, ANN, DTR, RFR (best); [18]: 25 ML methods (MLP best); [19]: BiLSTM+BO model	[15]: R2 = 0.92; MAE = 0.012; RMSE = 0.056; [18]: MAE = 0.006–0.009; R2 = 0.99; [19]: R2 ≈ 0.996–0.998; error rate <1.25%	Innovative ML application; high prediction accuracy; applicable to TPU-based orthosis

Although existing research has clarified the influence of certain process parameters on the performance of TPU printed parts, the systematic regulation laws governing the effect of process parameters on the flexibility of TPU printed parts remain to be improved. Additionally, the exploration of interactions between multiple parameters is insufficiently in-depth. Based on this, this paper employs elastic modulus as the evaluation index for flexibility. It conducts research through a combination of single-factor preliminary experiments and orthogonal experiments with interactions. The study deeply analyzes the comprehensive regulatory effects of each process parameter on the flexibility of TPU printed parts and introduces a binary polynomial model to fit and predict the elastic modulus of TPU printed parts, thereby optimizing the flexibility of TPU specimens. It is expected to provide reliable theoretical support and practical reference for the efficient optimization of FDM process parameters in TPU printing.

2. PRINTED MATERIALS AND METHODS

2.1. Printed material

A white TPU filament with a diameter of 1.75 mm, linear density of 1.21 g/cm³, Vicat softening point of 92°C, and Shore hardness of 95A was supplied by Zhongshan Tianrui 3D Printing Consumables Technology Co., Ltd.

2.2. Main equipment and instruments

The Bambu Lab A1 3D printer, equipped with a textured polyetherimide (PEI) build surface, was provided by Shenzhen Tuozhu Technology Co., Ltd.

The Shimadzu AGS-X series electronic universal testing machine was manufactured by Shimadzu Corporation, Japan.

2.3. Experimental methods

In this paper, two types of experiments—single-factor experiments and orthogonal experiments with interactions—were employed to analyze the influence of process parameters on flexibility (measured by elastic modulus). Single-factor experiments were conducted as preliminary studies, strictly adhering to the principle of controlled variables to investigate the influence mechanism of individual factors on mechanical properties. However, since single-factor experiments can only determine the individual influence of each process parameter, the combined effects and interactions of multiple factors on mechanical properties required verification through subsequent experiments. Based on the results of the single-factor experiments, an orthogonal experimental design with interactions was formulated in the subsequent phase to further detail the influence of process parameters on flexibility. The overall workflow is illustrated in Figure 1.

Figure 1
Flow chart of the research work.

3. TEST PROCESS AND RESULTS

3.1. Single-factor test procedures

3.1.1. Design of single-factor test scheme

Single-factor experiments were conducted to analyze the independent influence mechanisms of four process parameters—printing speed (A), printing temperature (B), infill rate (C), and layer height (D)—on the elastic modulus. In each test, only one factor was varied while the others were held constant. Based on the elastic modulus values obtained from the single-factor test results, the significance of each factor’s influence on the elastic modulus was determined.

The level design of the single-factor experiments for the four process parameters is presented in Table 2. The materials were printed and formed in accordance with the selected level values from Table 2.

Table 2
Single-factor variable experiment parameter settings.

ITEMS	FACTOR
	A/mm/s	B/℃	C/%	D/mm
Min	15	200	20	0.15
Max	55	240	80	0.35
Interval	10	10	15	0.05
Fixed value	35	210	80	0.2

3.1.2. Single-factor test model

To investigate the mechanical properties of TPU parts, experiments were conducted using Type 2A dumbbell standard specimens as the part model, in accordance with the requirements of the national standard

《 Determination of Tensile Stress-Strain Properties of Vulcanized Rubber or Thermoplastic Rubber 》(GB/T 528-2009). The model was three-dimensionally designed, converted into STL format, and imported into Bambu software for process parameter adjustment and slicing. The sliced files were transmitted to the 3D printer to execute the printing command according to the single-factor experimental scheme. After printing, the specimens were naturally cooled and removed after approximately 40 minutes. All printed parts were placed in the same environment for 2 days prior to tensile testing. Four specimens were printed for each single-factor level. The overall experimental process is shown in Figure 2.

Figure 2
The overall process of the experiments (a) Printing process (b) Molded part (c) Tensile process.

3.1.3. Single-factor test results and analysis

Because modifying a given factor produced highly similar stress-strain curves, a single representative curve is presented for each factor, and the stress-strain curves under single-factor conditions are shown in Figure 3. As observed in Figure 3, the stress-strain curves of TPU elastomers under tensile load exhibit the characteristic ‘S’-shaped nonlinear behavior unique to high polymer materials such as rubber. In the initial stage, when strain is less than 0.3, engineering stress increases linearly with strain, after which it transitions to a nonlinear stage. When strain exceeds 0.5, engineering stress increases significantly, indicating a distinct strain hardening phenomenon. Therefore, the results of 68 uniaxial tensile tests across 17 groups (duplicate specimens, represented by the row of repeated test conditions in the table, were excluded) were analyzed for strains in the range of 0–0.3. The elastic modulus of TPU specimens was obtained by fitting using the least squares fitting method, as shown in Table 3.

Figure 3
Stress-strain curves under single-factor conditions.

Table 3
Results of single-factor experiments.

VARIABLE FACTOR	LEVEL	A/(mm/s)	B/℃	C/%	D/mm	ELASTIC MODULUS/ MPA
						1	2	3	4
A	1	15	210	80	0.20	5.26	5.37	5.23	5.10
	2	25	210	80	0.20	5.81	5.73	5.91	5.76
	3	35	210	80	0.20	5.88	5.76	5.69	5.64
	4	45	210	80	0.20	5.95	5.66	5.49	5.74
	5	55	210	80	0.20	4.28	4.64	4.42	4.51
B	1	200	35	80	0.20	6.95	6.78	6.83	6.88
	2	220	35	80	0.20	5.63	6.06	5.70	5.45
	3	230	35	80	0.20	4.89	4.75	4.93	4.92
	4	240	35	80	0.20	4.64	4.80	4.74	4.79
C	1	35	210	20	0.20	4.76	4.60	4.39	4.48
	2	35	210	35	0.20	5.88	5.60	5.70	5.52
	3	35	210	50	0.20	5.85	6.12	5.71	5.96
	4	35	210	65	0.20	6.38	6.30	6.58	6.28
D	1	35	210	80	0.15	4.23	4.22	4.20	4.30
	2	35	210	80	0.25	4.50	4.35	4.41	4.36
	3	35	210	80	0.30	4.19	4.29	4.33	4.30
	4	35	210	80	0.35	4.23	4.41	4.28	4.22

To investigate the influence of factors A, B, C, and D on the flexibility of TPU specimens, the test data for the elastic modulus of TPU printed parts under various printing process parameters are presented in Table 4 in the form of mean ± standard deviation. The standard deviation for each group is less than 5%, indicating low intra-group dispersion and good repeatability of the test data. A homogeneity of variance test was conducted on the test data for the four factors. The results showed p-values of 0.595, 0.214, 0.865, and 0.571 for the four factors, respectively, all of which are greater than 0.05. This indicates that the test data for all factors meet the requirement of homogeneity of variance, satisfying the prerequisite for one-way analysis of variance (ANOVA). Subsequent ANOVA can therefore be performed on these data.

Table 4
Test data of elastic modulus for tpu printed parts under different process parameters.

LEVEL	ELASTIC MODULUS/ MPa
	A	B	C	D
1	5.24 ± 0.11b	6.86 ± 0.07a	4.56 ± 0.16d	4.24 ± 0.04c
2	5.80 ± 0.08a	5.74 ± 0.10b	5.68 ± 0.16c	5.74 ± 0.10a
3	5.74 ± 0.10a	5.71 ± 0.26b	5.91 ± 0.17b	4.41 ± 0.07b
4	5.71 ± 0.19a	4.87 ± 0.08c	6.39 ± 0.14a	4.28 ± 0.06c
5	4.46 ± 0.15c	4.74 ± 0.07c	5.74 ± 0.10bc	4.29 ± 0.08c

• Note: a represents the maximum value; the same letters indicate no significant difference between groups, while different letters indicate significant differences between groups.

The results of the one-way ANOVA indicated that factors A, B, C, and D all exerted significant influences on the flexibility of TPU specimens (p < 0.05). The Duncan’s multiple range test results were reliable, and the obtained specimen flexibility values are presented in Figure 4. Different letters above the columns in the figure denote significant differences between groups (p < 0.05).

Figure 4
Results of duncan’s multiple range test.

The influence of factor A (printing speed) on the elastic modulus exhibited a trend of an initial rise and subsequent decline. Printing speed directly impacts the thermal processing and flow characteristics of the filament within the nozzle. At high speeds, insufficient heating time for the filament results in inadequate melting and vulcanization, leading to loose interlayer bonding and reduced vulcanization degree. Conversely, at low speeds, the filament is fully heated and uniformly melted, promoting tight interlayer bonding, an appropriate vulcanization degree, good structural integrity, and formability of the specimens, while also avoiding issues such as uneven extrusion and nozzle clogging [²⁰]. Given that flexibility is the core objective of this study, a lower elastic modulus is more favorable for enhancing flexibility. Therefore, the low-speed and high-speed intervals of printing speed are intended for further exploration in subsequent orthogonal experiments. According to Duncan’s multiple range test, significant differences in elastic modulus were observed between groups 2/3/4 and group 5 and 1. However, no significant statistical differences were found among groups 2, 3, and 4. Based on the results of the significance difference analysis, three printing speed levels (15, 45, and 55 mm/s) that exhibit the smallest flexibility values and show significant differences (p < 0.05) were selected for the subsequent orthogonal experimental design.

The influence of printing temperature (B) on the elastic modulus exhibited a gradual decreasing trend. It affects the elastic modulus by regulating filament extrusion characteristics, bonding and stacking performance, and interlayer bonding force. At low temperatures, the high viscosity of the molten filament results in slow extrusion and weak interlayer bonding force. Conversely, at high temperatures, excessive filament fluidity may lead to over-melting, causing either excessive interlayer bonding or overflow, which tends to saturate and reduce the stability of interlayer bonding force. Therefore, the optimal high-temperature interval for printing temperature is intended to be further explored in subsequent orthogonal experiments [²¹]. According to Duncan’s multiple range test, significant differences in elastic modulus were observed between groups 2/3, groups 4/5, and group 1. However, no significant statistical differences were found among groups 2/3 and groups 4/5. Based on the results of significant difference analysis, three printing temperature levels (200, 220, and 240 ℃) with the smallest flexibility values and significant differences (p < 0.05) were selected for the subsequent orthogonal design.

The influence of the infill rate (C) on the elastic modulus exhibited a gradual increasing trend. As the infill rate increases, the filament consumption increases, and the filament arrangement becomes more compact. This leads to enhanced extrusion between adjacent filaments, improved interlayer bonding, and thus better mechanical properties, albeit with a significant reduction in flexibility [²²]. Consequently, the low infill rate interval is intended for further exploration in subsequent orthogonal experiments. According to Duncan’s multiple range test, significant differences in elastic modulus were observed between groups 3/5 and 2/5, as well as between group 1 and group 4. However, no statistically significant differences were found between groups 3/5 and 2/5. Based on the results of the significant difference analysis, three infill rate levels (20%, 35%, and 65%)—with the lowest flexibility values and significant differences (p < 0.05)—were selected for the subsequent orthogonal design.

The influence of layer height (D) on the elastic modulus exhibited a trend of initial increase followed by a decrease. When the layer height is small, although interlayer bonding is tight and delamination defects are absent, the reduced thickness layers results in insufficient overall rigid support of the part. This leads to strong stress dispersion between filaments, thereby reducing the elastic modulus. As layer height increases, the extrusion force exerted by the nozzle on the filament decreases, causing a reduction in interlayer bonding strength. This can result in poor bonding or even delamination, compromising the integrity of the part’s internal structure and consequently lowering the elastic modulus [²³]. Therefore, further exploration of lower and higher layer height intervals is planned for subsequent orthogonal experiments. According to Duncan’s multiple range test, significant differences in elastic modulus were observed between groups 1/4/5 and group 2, as well as between groups 1/4/5 and group 3. However, no statistically significant differences were found among groups 1, 4, and 5. Based on the results of the significant difference analysis, three layer height levels (0.2, 0.25, and 0.3 mm) with the smallest flexibility values and significant differences (p < 0.05) were selected for the subsequent orthogonal design.

3.2. Orthogonal experiments

3.2.1. Determination of orthogonal experiment levels

Based on the sensitivity of the four process parameters to flexibility observed in factor experiments, the statistically optimal levels for each significant single factor were determined using Duncan’s multiple range test. The selected level values were factor A (15, 45, 55 mm/s), factor B (200, 220, 240 ℃), factor C (20, 35, 65%), and factor D (0.2, 0.25, 0.3 mm).

3.2.2. Orthogonal experiment design scheme and results

Based on these results, orthogonal experiments with interactions were carried out, constructing a total of 27 groups of experiments with four factors and three levels, and the design and results of the orthogonal experiments are shown in Table 5. The part models were printed according to the experimental conditions, and 3 specimens were printed for each experiment number; after printing, all specimens were placed at room temperature for 24 hours. The final test results were obtained in accordance with the requirements.

Table 5
Orthogonal test scheme with interaction effects and results.

No.	A	B	A × B(2COLS)	C	A × C(2COLS)	B × C(2 COLS)	D	ERROR(2 COLS)	ELASTICMODLUS/MPA
									1	2	3
1	1	1		1			1		6.85	6.35	6.37
2	1	1		1			2		6.30	6.85	6.23
3	1	1		1			3		6.29	6.34	6.33
4	1	2		2			1		6.57	6.53	6.53
5	1	2		2			2		6.62	6.63	6.62
6	1	2		2			3		6.45	6.49	6.49
7	1	3		3			1		6.85	6.83	6.85
8	1	3		3			2		6.84	6.85	6.82
9	1	3		3			3		6.83	6.84	6.83
10	2	1		2			1		5.86	5.86	5.86
11	2	1		2			2		5.88	5.91	5.86
12	2	1		2			3		5.89	5.88	5.88
13	2	2		3			1		6.16	6.17	6.19
14	2	2		3			2		6.35	6.36	6.38
15	2	2		3			3		6.35	6.37	6.36
16	2	3		1			1		6.85	6.35	6.35
17	2	3		1			2		6.26	6.32	6.35
18	2	3		1			3		6.58	6.39	6.39
19	3	1		3			1		5.88	5.88	5.88
20	3	1		3			2		5.86	5.86	5.88
21	3	1		3			3		5.88	5.88	5.88
22	3	2		1			1		5.90	5.87	6.12
23	3	2		1			2		6.09	6.12	6.18
24	3	2		1			3		6.19	6.19	5.87
25	3	3		2			1		6.44	6.40	6.39
26	3	3		2			2		6.60	6.35	6.36
27	3	3		2			3		6.38	6.36	6.37

• Note: 1, 2, 3 indicate different factor levels. Interaction columns (A × B, A × C, B × C) are used for variance analysis and not shown with level numbers in this table for brevity.

4. ANALYSIS OF TEST RESULTS

4.1. Analysis of variance (ANOVA)

A variance analysis was conducted on the model, as presented in Table 6. The influence degree and significance of process parameters on the elastic modulus were determined by integrating F values and p values. To more clearly illustrate the significant combinations, non-significant variance sources were excluded from Table 6.

Table 6
ANOVA and significance test for elastic modulus.

SOURCE OF VARIATION	SUM OF SQUARES	DEGREE OF FREEDOM	MEAN SQUARE	F-VALUE	P-VALUE
Model**	2.438	8	0.305	56.316	p < 0.01
A**	1.239	2	0.619	114.483	p < 0.01
B**	1.106	2	0.553	102.168	p < 0.01
AB**	0.093	4	0.023	4.307	0.013
Residual	0.097	18	0.005
Cor Total	2.535	26
R2 = 0.962$R_{\text{Adj}}^2$ = 0.945

• Note:** — indicates that the factor has an extremely significant impact on the experimental results.

The model probability (p < 0.01) in Table 6 indicates that the orthogonal experiment model is highly significant. The comprehensive influence of the selected process parameters on the TPU elastic modulus is also highly significant, and the experimental design demonstrates clear significance. The coefficient of determination (R²) of the model is 0.962, and the adjusted coefficient of determination (Adj R²) is 0.945. These values indicate that the model exhibits a good fit to the experimental data, effectively reflecting the variation relationship between process parameters and response values. Consequently, the selected factors and experimental data can be directly utilized for the construction of subsequent linear regression models and the optimization analysis of process parameters. The factors with significant influence on the TPU elastic modulus (p < 0.05) are printing speed (A), printing temperature (B), and their interaction term (A × B). The order of significance from highest to lowest is B > A > AB.

4.2. Regression analysis

A quadratic linear polynomial regression model was established, with two significant single factors and one significant interaction term as independent variables and the TPU elastic modulus as the response value. To eliminate the influence of differences in order of magnitude and dimension between process parameters and their interaction terms on the fitting accuracy of the model, the significant factors (printing speed x_₁, printing temperature x_₂) and response value (TPU tensile Young’s modulus Y) selected from the orthogonal experiments were preprocessed using the [–1,1] interval normalization method. The normalization formula (1) is given as follows [²⁴]:

(1) $$x_{\text{i}}^{\prime }=\frac{2(x-x_{min})}{x_{max}-x_{min}}-1$$

where x_min represents the minimum value of each process parameter, x_max the maximum value, x the value of the process parameter to be normalized, and x_ᵢ′ the normalized value of x.

Given the significant interaction between printing speed and printing temperature (x_₁× x_₂, p < 0.05), a quadratic nonlinear model incorporating the normalized interaction term was established as follows:

(2) $$Y\text{'}=-0.89-0.494x_{\text{1}}^{\prime }+0.482x_{\text{2}}^{\prime }+0.072x_{\text{1}}^{\prime 2}-0.094x_{\text{2}}^{\prime 2}-0.016x_{\text{1}}^{\prime }{x}_{\text{2}}^{\prime }$$

where Y′ is the normalized Young’s modulus, $x_{\text{1}}^{\prime }$ is the normalized process parameter, and $x_{\text{1}}^{\prime }{x}_{\text{2}}^{\prime }$ is the normalized interaction term of $x_{\text{1}}^{\prime }$ and $x_{\text{2}}^{\prime }$.

A normality test was conducted on the residuals of the quadratic polynomial regression model, with the residual Q-Q (Quantile-Quantile) plot presented in Figure 5. As observed in the figure, all data points lie within the 95% confidence interval, with only a few exhibiting slight deviations and no significant anomalies, generally aligning with the reference diagonal line. This indicates that the model residuals follow a normal distribution, and the unexplained errors are attributable to experimental random variation. Consequently, the model satisfies the fundamental statistical assumptions of regression analysis [²⁵].

Figure 5
Residual distribution plot of the elastic modulus regression model.

To verify the reliability of the established quadratic polynomial regression model for TPU elastic modulus, analysis of variance (ANOVA) and significance tests were conducted, as presented in Table 7. The results indicate that the model is extremely significant (p < 0.001), confirming that the regulatory effect of FDM process parameters on TPU elastic modulus is statistically significant and that interference from accidental experimental factors can be excluded. The lack of fit term (p > 0.05) is not significant, suggesting the model has no structural defects and that unexplained residuals arise solely from experimental random errors—thereby verifying that the quadratic polynomial effectively fits the nonlinear correlation between process parameters and TPU elastic modulus. The model’s coefficient of determination (R²) is 0.850, explaining 85% of the variation in elastic modulus. The adjusted coefficient of determination (Adj R² = 0.840) is highly close to R², indicating the absence of redundant variables and no overfitting. The predicted coefficient of determination (Pred R² = 0.848) is slightly lower than R², demonstrating excellent generalization ability. These findings collectively confirm the model’s reliability, effectiveness, and rationality, providing a mathematical tool for regulating TPU elastic modulus via FDM process parameters and establishing a statistical foundation for subsequent process optimization.

Table 7
Analysis of variance and significance test for quadratic linear polynomial.

SOURCE	SUM OF SQUARES	df	MEAN SQUARE	F-VALUE	P-VALUE	STATISTICAL SIGNIFICANCE
Regression	28.353	5	5.671	84.082	0.000	significant
Residual	5.058	75	0.067
Lack of fit	2.071	21	0.099	1.782	0.068	Not significant
Pure error·	2.987	54
Cor total	33.411	80
R2 = 0.850 $R_{\text{Adj}}^2$ = 0.840 Pred R2 = 0.848

4.3. Objective optimization

4.3.1. Extreme value analysis

An extreme value analysis was conducted on the established quadratic polynomial regression equation. By calculating the first-order partial derivatives and setting them equal to zero (∂Y'/∂X'₁ = 0, ∂Y'/∂X'₂), the stationary point coordinates were determined as (x₁′ ≈ 4.23, x₂′ ≈ 2.34). Furthermore, the discriminant Δ ≈ −0.027 < 0 was calculated using the second-order partial derivatives, indicating that this stationary point is a saddle point and the regression equation has no global minimum value. Therefore, in conjunction with the intuitiveness of the 3D surface plot, the feasible minimum value was identified within the reasonable parameter range.

4.3.2. Interaction effect analysis

To further investigate the influence of the significant interaction between factors A and B on the elastic modulus and determine the feasible minimum value within the reasonable parameter range, the interaction effect law of the two factors is presented in Figure 6. When factor A is within the interval (15, 55) mm/s and factor B is within the interval (200, 220) °C, the elastic modulus is larger under conditions of slower printing speed and higher printing temperature, and smaller under opposite conditions. Therefore, to study the flexibility of TPU materials, factors A = 55 mm/s and B = 200 °C were selected for synergistic regulation, as this combination resulted in the smallest material flexibility, thereby being identified as the optimal process parameters.

Figure 6
Interaction law of factors A and B (a) Contour plot for printing speed and printing temperature (b) 3D surface plot for printing speed and printing temperature.

4.3.3. Test verification

To verify the practical feasibility of the optimal process parameters (A = 55 mm/s and B = 200 °C), five parallel printed and mechanically tested specimens were fabricated under identical test conditions. The measured TPU elastic modulus values were 5.875, 5.88, 5.88, 5.86, and 5.825 MPa. The coefficient of variation (CV) was employed to assess the stability and reproducibility of the test results. The formula for the CV is as follows:

(3) $$\begin{array}{l}cv\%=\frac{s}{\overline{x}}\times 100\%\\ s=\sqrt{\frac{{\displaystyle \sum _{i=1}^n{(x_i-\overline{x})}^2}}{n-1}}\\ \overline{x}=\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}\end{array}$$

where x– is the sample mean; s is the sample standard deviation; n is the sample size; and x_i is the individual measured value.

Verification results confirm that the identified set of process parameters represents the true optimal solution. Five groups of parallel experiments were conducted under the parameters of a printing speed of 55 mm/s and a printing temperature of 200 °C. The coefficient of variation of the measured elastic modulus was 0.40% (< 5%), demonstrating stable and reproducible results. Concurrently, the hot-melt bonding behavior of FDM-printed TPU under these parameters aligns with the variation pattern of the elastic modulus, thereby achieving the integration of the optimal model and process mechanism.Based on FDM molding mechanisms and TPU material characteristics, 200 °C falls within the generally accepted suitable melting interval for TPU. This temperature theoretically facilitates complete material melting without thermal degradation, ensuring the integrity of molecular chains and appropriate viscous flow fluidity, thus providing theoretical support for effective interlayer bonding [²⁶]. The printing speed of 55 mm/s is compatible with the melt deformability of TPU at this temperature, which theoretically effectively mitigates issues such as nozzle heat accumulation and lagging filament extrusion rate. This ensures uniform filament spreading and stability of the interlayer contact area [²⁷]. These findings are also consistent with relevant research conclusions on optimizing TPU printing parameters via orthogonal experiments, thereby effectively validating the accuracy and reliability of the optimal process parameters for elastic modulus obtained through orthogonal experiments.

5. CONCLUSIONS

The primary motivation of the proposed systematic optimization method is to address critical challenges in existing FDM process parameter optimization for Thermoplastic Polyurethane (TPU) parts. These challenges include the lack of systematic parameter level screening, insufficient quantitative analysis of parameter interaction priority, and inadequate rigorous stability verification of optimal parameters. Compared with other existing approaches, the proposed method offers three distinct advantages: pre-screening parameter levels to enhance experimental efficiency, quantifying the priority of parameter interactions to clarify their intrinsic influence on TPU flexibility, and conducting rigorous stability verification of optimal parameters to ensure their reliability and industrial applicability.

To clarify the influence rules, optimal combination, and selection principles of the process parameters, this study employed SPSS statistical analysis, single-factor analysis of variance (ANOVA), Duncan’s multiple range test, orthogonal experiments, and verification experiments. The original aspects of this study and its differences from other relevant studies are clearly reflected in the following three aspects:First, unlike most existing studies that directly set parameter levels for orthogonal experiments (which can easily lead to redundant levels and low experimental efficiency), this study uses single-factor ANOVA and Duncan’s multiple range test to pre-screen significant difference levels; Second, compared with studies that focus solely on the impact of single parameters on mechanical properties, this study quantifies the priority of the influence of parameters and their interaction terms on the elastic modulus. Third, unlike studies that lack stability verification of optimal parameters, this study ensures the reproducibility of optimal parameters through rigorous parallel testing, thereby enhancing the applicability of the research results to industrial production.The specific conclusions are as follows:

1. Through single-factor ANOVA and Duncan’s multiple range test, three significant difference levels for each of the four process parameters were successfully identified. Group differences can be intuitively distinguished via visual charts. This provides a scientific basis for the rational design of subsequent orthogonal experiments, effectively reducing redundancy in factor levels and improving experimental efficiency. This represents a key distinction from many traditional studies, which often overlook pre-screening of parameter levels, potentially leading to inefficient use of experimental resources.

2. The effects of process parameters on the flexibility and elastic modulus of TPU specimens exhibit clear regularity. Single-factor ANOVA confirms that all four factors—printing speed, printing temperature, infill rate, and layer height—exert an extremely significant impact on the flexibility of TPU specimens (p < 0.001), findings consistent with those of existing relevant studies. However, this study further clarifies, through orthogonal experiment analysis, the priority of factors and their interaction terms affecting elastic modulus: printing temperature (B) > printing speed (A) > A × B interaction term (p < 0.05). This fills the gap left by existing studies, which have failed to quantify the priority of parameter interaction effects. Additionally, the influence trends of each factor on elastic modulus differ: the effects of printing speed and layer height on elastic modulus initially increase and then decrease; the effect of printing temperature decreases gradually; and the effect of infill rate increases gradually. These insights provide specific theoretical and experimental foundations for the precise regulation of process parameters.

3. With the optimal flexibility of TPU specimens as the core objective, orthogonal experiments and verification experiments were conducted to determine the optimal process parameters: printing speed A = 55 mm/s and printing temperature B = 200 ℃. Concurrently, targeted selection principles for process parameters were proposed, specifying the optimal range for each parameter and the forming defects to be avoided. Unlike existing studies that lack stability verification of optimal parameters, this study ensures that the coefficient of variation (CV) remains ≤ 5% when the optimal parameters are applied in parallel tests (actual veriffcation CV = 0.40%). This guarantees the stable reproducibility of forming results and provides a practical technical reference for the industrial production of FDM flexible parts using TPU materials.

Despite the aforementioned research achievements, this study presents certain limitations that require attention in future work. First, the study focuses exclusively on four core process parameters and does not account for secondary process parameters such as printing environment humidity, nozzle diameter, and cooling speed, which may impact the generalizability of the research conclusions. Second, experiments were conducted using a single type of TPU filament, and the applicability of the optimized process parameters to other TPU material types remains to be further validated. Third, the proposed optimization scheme is primarily suitable for small-scale FDM equipment, and its adaptability to large-scale industrial FDM systems requires further exploration.

Based on the aforementioned limitations, future research will focus on three key aspects: first, expanding the range of TPU wire models, optimizing process parameters in accordance with the characteristics of different materials, and enhancing the universality of research conclusions; second, incorporating additional secondary process parameters, analyzing their interactions with core parameters, and refining the process parameter regulation system; third, optimizing process parameters and forming schemes for large-scale industrial FDM equipment, addressing the stability challenges of batch production in industrial settings, and further advancing the industrial application of TPU material-based FDM flexible parts. Additionally, future research will integrate the entire life cycle of the printing process to develop a sustainable and environmentally friendly FDM process, while increasing the volume of available data to further improve the accuracy of mechanical property prediction and parameter optimization.

6. Acknowledgments

This work is supported by the Key Natural Science Research Project of Bengbu University under Grant No.2023ZR05zd.

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Publication Dates

• Publication in this collection
  05 June 2026
• Date of issue
  2026

History

• Received
  17 Feb 2026
• Accepted
  13 Apr 2026