Targeted optimization of flatness accuracy for FDM 3D printing using PLA

ABSTRACT

Planar accuracy is a critical geometric precision requirement for components, directly influencing their functional performance. To investigate the relationship between process parameters and planar accuracy in the open Fused Deposition Modeling (FDM) process, and to enhance the shape accuracy of printed models, a series of experiments, including Plackett-Burman (PB) and Response Surface Methodology (RSM) experiments, were conducted. The experimental data were then used to develop nonlinear regression models for optimization. Eight process parameters—nozzle temperature, layer thickness, substrate temperature, external perimeter speed, line width, extrusion flow ratio, fill density, and wall layer count—were initially selected, with each parameter tested at two levels (high and low). PB experiments were conducted, and the results revealed that nozzle temperature, external perimeter speed, extrusion flow ratio, and wall layer count significantly affected the flatness error of the printed parts. Based on these findings, a Central Composite Design (CCD) was employed. The results indicated that nozzle temperature (A), external perimeter speed (B), extrusion flow rate (C), and wall layer count (D) were the primary factors influencing the surface flatness of the models. Moreover, significant interaction effects were observed between AC, AD, BC, BD, and CD. Subsequently, nonlinear regression models for the surface flatness were derived using Design-Expert software. MATLAB was used for objective optimization, leading to the optimal parameters: A = 230°C, B = 60 mm/min, C = 0.96, and D = 2. Under these conditions, the signal-to-noise ratio for flatness error reached its maximum value of 42.247. Finally, a model was printed using the optimized parameters, and the results validated the optimization process. This study provides valuable theoretical insights into improving the shape accuracy of FDM-printed PLA components.

Keywords:
PLA; FDM; PB experiment; RSM-CCD experiment

1. INTRODUCTION

Fused Deposition Modeling (FDM) is a widely used additive manufacturing technique, known for its simplicity and cost-effectiveness [¹, ²]. With continuous advancements in FDM technology and the upgrading of printing equipment, significant changes have occurred in the printing process. These advancements include accelerated increases in print speed and enhanced capabilities for adjusting multiple process parameters [³], which have made the overall combination of process parameters more flexible and versatile. As most 3D printers operate with an open-loop control system, the printing parameters must be set prior to the start of the additive manufacturing process and cannot be modified during printing [⁴]. This requirement forces operators to identify the optimal parameters in advance from a wide range of adjustable settings, ensuring they are tailored to the specific application scenarios and performance requirements of the printed parts.

Current research on printed parts has primarily concentrated on dimensional accuracy and mechanical properties. For example, PAWAR and DOLAS [⁵] explored the relationship between process parameters, bending strength, and surface roughness in PC-ABS parts, concluding that a layer thickness of 0.14 mm and 100% infill density yielded the highest strength and optimal surface quality. Similarly, PATIL et al. [⁶] employed Response Surface Methodology (RSM) to analyze the effects of air gap, raster angle, and build orientation on bending performance. WANG et al. [⁷] conducted orthogonal experiments to evaluate the influence of various process parameters on dimensional accuracy, identifying layer thickness as the most influential factor. ZHAO and SHAO [⁸] developed a machine learning-based coupled model to predict the surface roughness of Fused Deposition Modeling (FDM) parts, highlighting layer thickness, infill density, nozzle temperature, and external perimeter speed as the dominant contributing factors. Furthermore, ZHOU and ZHANG [⁹] studied the optimization of molding quality for nylon filaments and found that layer thickness, infill density, and nozzle temperature significantly affect both dimensional accuracy and tensile strength.

However, in practical applications, the shape accuracy of printed parts—encompassing parameters such as flatness, roundness, and cylindricity—is equally critical. Relying on post-processing techniques to enhance shape accuracy not only increases production costs and processing time but may also compromise overall dimensional integrity. Therefore, improving shape accuracy directly during the printing process is of significant practical importance.

This paper aims to identify the adjustable parameters that influence flatness accuracy among various process variables, assess their significance, and explore their relationship with flatness error. The objective is to optimize these parameters to minimize flatness error. Since flatness error is influenced by both dimensional error and surface roughness, which are crucial to part quality and accuracy, any adjustable process parameters affecting dimensional accuracy or surface roughness are considered potential factors for investigation in this study.

2. PRINTED MATERIALS AND METHODS

2.1. Printed material

When selecting printing materials, commonly used options are chosen to enhance the generalizability of the results. Polylactic acid (PLA) is a widely employed material in Fused Deposition Modeling (FDM) printing, known for its biodegradability and absence of irritating odors during the printing process. It is one of the most popular materials in FDM technology [¹⁰, ¹¹]. PLA-based prints exhibit low shrinkage, high dimensional stability, excellent mechanical properties, and minimal warping and deformation [¹²,¹³,¹⁴]. In this study, PLA Basic, with a filament diameter of 1.75 mm and a density of 1.24 g/cm³, was used. The PLA was sourced from Shenzhen Top Bamboo Co.

2.2. Equipment and instruments

The printing equipment used was the Bambu Lab P1S (Shenzhen Top Bamboo Co., Ltd.), while the measurement equipment employed was the Hexagon CMM, equipped with PC-DMIS PRO 2010 software.

2.3. Experimental methods

This study employs two experimental methods: the Plackett-Burman (PB) design and the Response Surface Methodology with Central Composite Design (RSM-CCD), to investigate the influence of process parameters on the flatness of the model.

A large number of process parameters can be adjusted in an open-ended manner. In the early stage, the Plackett–Burman (PB) design was employed to investigate whether multiple factors influence flatness accuracy. The primary advantage of the PB design is that, when dealing with numerous factors, it substantially reduces the number of experiments compared with orthogonal, response surface, or factorial designs, thereby improving efficiency. However, because the PB design involves relatively few runs, the evaluation may be incomplete and sometimes yield false positives. Therefore, subsequent validation experiments are required to enhance the reliability of the results. Based on the PB results, the RSM-CCD design will be used to further explore the detailed effects of process parameters on flatness deviation. The overall experimental procedure is shown in Figure 1.

Figure 1
Workflow diagram.

3. EXPERIMENTAL PROCESS AND RESULTS

3.1. PB experimental procedure

3.1.1. PB experimental design

To identify the key factors influencing the results from a large set of process parameters, improve experimental efficiency, and minimize the number of trials, a Plackett-Burman (PB) design was employed. Eight factors were selected: nozzle temperature (A), layer thickness (B), substrate temperature (C), external perimeter speed (D), line width (E), extrusion flow ratio (F), fill density (G), and wall layer count (H). Each factor was assigned high and low levels, leading to a 2-level, 8-factor design, as summarized in Table 1.

Table 1
PB experimental factor levels table.

LEVELS	A	B	C	D	E	F	G	H
–1	190°C	0.08 mm	60°C	60 mm/s	0.3 mm	0.96	15%	2
+1	230°C	0.28 mm	70°C	200 mm/s	0.7 mm	1.04	50%	6

3.1.2. PB experiment model printing

The experimental model was a cube with dimensions of 25 mm × 25 mm × 25 mm. After finalizing the design, the model was converted to the STP format and imported into Bamboo Studio software, where the print command was executed according to a predefined experimental protocol. Upon completion of printing, the model was allowed to cool naturally within the enclosed chamber for approximately 40 minutes before removal. Once all prints were finished, they were stored under the same environmental conditions for 48 hours prior to measurements. This is to ensure that the measured differences more reliably reflect the influence of printing parameters, rather than being affected by stress release or moisture changes.The final printed model is shown in Figure 2.

Figure 2
Modeling of PB experiments.

3.1.3. Analysis of PB experiment results

The flatness errors of the top and side surfaces of the twelve models were measured using a coordinate measuring machine (CMM). For each surface, nine points were selected from the most distant locations, as shown in Figure 3.

Figure 3
Location of collection points.

These points were used to fit a reference plane, from which the corresponding flatness errors were calculated. The measurement results are summarized in Table 2.

Table 2
Results of PB experiments.

STD	A	B	C	D	E	F	G	H	TOP SURFACE FLATNESS ERROR	SIDE FLATNESS ERROR
1	+1	+1	–1	+1	+1	+1	–1	–1	0.053 mm	0.052 mm
2	–1	+1	+1	–1	+1	+1	+1	–1	0.050 mm	0.054 mm
3	+1	–1	+1	+1	–1	+1	+1	+1	0.048 mm	0.036 mm
4	–1	+1	–1	+1	+1	–1	+1	+1	0.031 mm	0.029 mm
5	–1	–1	+1	–1	+1	+1	–1	+1	0.037 mm	0.065 mm
6	–1	–1	–1	+1	–1	+1	+1	–1	0.073 mm	0.090 mm
7	+1	–1	–1	–1	+1	–1	+1	+1	0.012 mm	0.011 mm
8	+1	+1	–1	–1	–1	+1	–1	+1	0.027 mm	0.032 mm
9	+1	+1	+1	–1	–1	–1	+1	–1	0.021 mm	0.021 mm
10	–1	+1	+1	+1	–1	–1	–1	+1	0.034 mm	0.038 mm
11	+1	–1	+1	+1	+1	–1	–1	–1	0.028 mm	0.027 mm
12	–1	–1	–1	–1	–1	–1	–1	–1	0.021 mm	0.053 mm

Comparison of the top and side flatness errors of the 12 models reveals that the overall trend in the errors is consistent, as shown in Figure 4. The only notable exception is Model 12, which exhibits a pronounced deviation. This deviation could be explained by the fact that the accuracy of the side surface strongly depends on interlayer bonding strength and internal stress levels. These factors are particularly sensitive to low-level parameter settings, which may induce interlayer misalignment and weakened adhesion on the side surfaces. Consequently, the flatness error of the side surface is often greater than that of the top surface. Overall, however, the flatness errors of the top and side surfaces remain comparable across most models.

Figure 4
Top and side flatness error.

Variance analysis (ANOVA) was performed on the top and side flatness, and the results are shown in Tables 3 and 4.

Table 3
Top surface flatness error ANOVA table.

SOURCE	SUM OF SQUARES	DF	MEANSQUARE	FVALUE	P-VALUEPROB>F
Model	0.0216	8	0.0027	66.26	0.0027	significant
A	0.0020	1	0.0020	49.82	0.0058
B	0.0000	1	0.0000	0.8255	0.4305
C	0.0001	1	0.0001	2.71	0.1980
D	0.0060	1	0.0060	146.43	0.0012
E	0.0001	1	0.0001	1.68	0.2854
F	0.0116	1	0.0116	283.82	0.0005
G	0.0003	1	0.0003	7.33	0.0733
H	0.0015	1	0.0015	37.48	0.0088
Residual	0.0001	3	0.0000
Total	0.0217	11
R2 = 0.9944 RAdj = 0.9794

Table 4
Side flatness error ANOVA table.

SOURCE	SUM OF SQUARES	DF	MEANSQUARE	FVALUE	Q-VALUEPROB>F
Model	0.0294	8	0.0037	19.24	0.0168	significant
A	0.0115	1	0.0115	60.13	0.0045
B	0.0007	1	0.0007	3.74	0.1488
C	0.0001	1	0.0001	0.54	0.5160
D	0.0008	1	0.0008	4.15	01343
E	0.0005	1	0.0005	2.74	0.1964
F	0.0114	1	0.0114	59.63	0.0045
G	0.0010	1	0.0010	5.00	0.1114
H	0.0034	1	0.0034	17.99	0.0240
Residual	0.0006	3	0.0002
Total	0.0300	11
R2 = 0.9809 R2Adj = 0.9299

As shown in Table 3, the F-value for the top surface flatness error model is 66.26, with a significance probability (p) of 0.0027, indicating high statistical significance. The p-values for the main factors A, D, F, and H are all below 0.01, suggesting a highly significant effect of these factors on the top surface flatness. In Table 4, the F-value for the side surface flatness error model is 19.24, with a significance probability of 0.0168, confirming its significance. For this model, the p-values for factors A and F are both below 0.01, indicating a highly significant influence on side surface flatness. Additionally, the p-value for factor H is 0.024, which is below 0.05, signifying a significant effect of H on the side surface flatness. The R² values for the two models are 0.9944 and 0.9809, respectively, with adjusted R² values of 0.9794 and 0.9299, reflecting a strong model fit and explaining over 92% of the variance in the effect sizes [¹⁵].

3.2. RSM-CCD experiment

Since the PB experiment cannot distinguish between the effects of main factors and interactions, its primary objective is to identify the factors that significantly influence flatness error among numerous variables. As a result, the PB experiment primarily provides key information for subsequent studies [¹⁶]. Building upon this, the RSM-CCD experiment is designed with orthogonality, rotationality, and sequentiality, enabling a systematic evaluation of parameter effects across various combinations of levels [¹⁷].

3.2.1. CCD experimental factor levels

A CCD experimental design was developed based on the results of the PB experiment. Of the eight factors, the significant ones—namely nozzle temperature (A), external perimeter speed (B), extrusion flow rate (C), and wall layer Count (D)—were selected for further investigation. The number of zero-level trials (m₀) was set to 6. Thus, based on the number of primary factors and the zero-level trials, the star point value (λ) for the CCD experiment was determined to be 1.72 [¹⁸]. The experimental factor levels are presented in Table 5.

Table 5
Table of factor levels for the CCD experiment.

LEVELS	A	B	C	D
–λ	190°C	60 mm/s	0.960	2
–1	198°C	89 mm/s	0.977	3
0	210°C	130 mm/s	1.000	4
+1	222°C	171 mm/s	1.023	5
+λ	230°C	200 mm/s	1.040	6

3.2.2. CCD experiment results

A total of 30 experiments were performed, necessitating the fabrication of 30 models, as shown in Figure 5.

Figure 5
Printed model of CCD experiment.

Since the PB experiments revealed consistent trends in the top and side surface flatness errors across different models, the response variable in the CCD experiments was selected as the top surface flatness error to streamline the measurement process. The top surface flatness error was measured five times for each model. To minimize experimental errors, the signal-to-noise ratio (S/N) was introduced as a metric to quantify the magnitude of fluctuations in the stability of output characteristics during orthogonal experiments [¹⁹]. Data analysis and processing were conducted using the S/N ratio corresponding to each experimental result. This approach not only enabled the analysis of controllable factors but also mitigated the influence of random interference, thereby enhancing the accuracy of the computational results [²⁰]. Given that flatness errors exhibit the “smaller-the- better” characteristic, with an ideal value of 0, the S/N ratio formula based on this characteristic was employed for calculation, as shown in Equation 1. A larger S/N ratio indicates a smaller surface flatness error, thus reflecting better precision.

(1) $$\raisebox{1ex}{$S$}\!\left/ \!\raisebox{-1ex}{$N$}\right.\text{=}-1\text{0}*lg(\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{y}_i^2})$$

In the above equation, n represents the number of measurements taken for the surface flatness error (set to 5). y_i denotes the measurement result of the iii-th reading, and S/N refers to the signal-to-noise ratio, which corresponds to the final response value in the CCD experiment (Table 6).

Table 6
CCD experimental program and results.

STD	A	B	C	D	S/N FOR FLATNESS ERROR
1	–1	–1	–1	–1	33.737
2	1	–1	–1	–1	35.453
3	–1	1	–1	–1	31.721
4	1	1	–1	–1	32.305
5	–1	–1	1	–1	31.440
6	1	–1	1	–1	34.541
7	–1	1	1	–1	31.058
8	1	1	1	–1	32.872
9	–1	–1	–1	1	31.821
10	1	–1	–1	1	32.381
11	–1	1	–1	1	31.499
12	1	1	–1	1	31.412
13	–1	–1	1	1	29.102
14	1	–1	1	1	30.318
15	–1	1	1	1	29.235
16	1	1	1	1	30.204
17	–1.72	0	0	0	29.028
18	1.72	0	0	0	32.487
19	0	–1.72	0	0	34.071
20	0	1.72	0	0	31.012
21	0	0	–1.72	0	31.347
22	0	0	1.72	0	29.225
23	0	0	0	–1.72	33.030
24	0	0	0	1.72	29.818
25	0	0	0	0	30.604
26	0	0	0	0	28.903
27	0	0	0	0	30.297
28	0	0	0	0	29.866
29	0	0	0	0	30.224
30	0	0	0	0	30.516

4. ANALYSIS OF EXPERIMENTAL RESULTS

4.1. ANOVA

ANOVA was performed on the response variables, and a quadratic fitting model was applied based on the p-values, as shown in Table 7. The model yielded an F-value of 22.94, with a p-value less than 0.0001, indicating that the model is highly statistically significant. The p-values for the main factors (A, B, C, and D) were all below 0.001, suggesting that these factors significantly impact the top surface flatness. The p-values for the interaction terms (AC, AD, BC, BD, and CD) were all less than 0.05, indicating significant interactions affecting the flatness error. Additionally, the quadratic terms (A², B², and D²) demonstrated a substantial effect on the flatness error.

Table 7
Top surface flatness error ANOVA table.

SOURCE	SUM OF SQUARES	DF	MEANSQUARE	F VALUE	R-VALUEPROB>F
Model	81.27	14	5.80	22.94	<0.0001	significant
A-A	11.42	1	11.42	45.14	<0.0001
B-B	8.62	1	8.62	34.08	<0.0001
C-C	10.55	1	10.55	41.71	<0.0001
D-D	23.47	1	23.47	92.76	<0.0001
AB	0.69	1	0.69	2.71	0.1204
AC	1.17	1	1.17	4.62	0.0482
AD	1.30	1	1.30	5.13	0.0388
BC	1.22	1	1.22	4.83	0.0440
BD	2.21	1	2.21	8.72	0.0099
CD	1.53	1	1.53	6.05	0.0265
A2	1.27	1	1.27	5.02	0.0406
B2	13.32	1	13.32	52.65	<0.0001
C2	0.21	1	0.21	0.84	0.3748
D2	4.28	1	4.28	16.93	0.0009
Residual	3.80	15	0.25
Lack of fit	1.83	10	0.18	0.47	0.8571	not significant
Pure Error	1.96	5	0.39
Cor Total	85.07	29
R2 = 0.9554 R2Adj = 0.9137

4.2. Main effects analysis

The relationship between the four main factors — nozzle temperature (A), external perimeter speed (B), extrusion flow rate ratio (C), and wall layer count (D) — and the signal-to-noise ratio (SNR) was analyzed using Design-Expert software (Figure 6).

Figure 6
Main effects analysis chart: (a) Nozzle temperature level and signal-to-noise; (b) External perimeter speed level and signal-to-noise; (c) Extrusion flow rate level and signal-to-noise; (d)Wall layer count level and signal-to-noise.

When all other factors were set to their zero levels, an increase in nozzle temperature (A) was found to raise the SNR, suggesting that, within a certain range, higher nozzle temperatures enhance flatness accuracy. This improvement is attributed to increased material fluidity, which facilitates better adhesion of the extruded filament to the printing surface, thereby mitigating issues such as uneven surfaces due to insufficient fusion and improving bonding strength between layers. Additionally, interlayer constraints are reduced, limiting material deformation.

In contrast, as external perimeter speed (B) increased from low to high levels, the SNR decreased, indicating that higher printing speeds lead to reduced flatness accuracy. This is likely due to the insufficient spreading of the melt, which cools too quickly during faster printing, weakening the bond strength between layers and diminishing interlayer constraints, thereby increasing material deformation.

For the extrusion flow rate ratio (C), increasing the ratio from low to high levels resulted in a decrease in the SNR. This suggests that a higher extrusion flow rate causes material accumulation, with excess material failing to be effectively scraped off during nozzle movement, leading to surface ripples.

Lastly, as wall layer count (D) increased from low to high levels, the SNR also decreased. Although additional layers can help cover the infill pattern and reduce surface texture, excessive layers result in edge accumulation, forming “convex edges” that reduce flatness. Moreover, small errors in each layer accumulate at higher layer counts, exacerbating the overall unevenness of the facade.

4.3. Interaction effects analysis

To thoroughly investigate the impact of two-factor interactions on flatness error, relevant conclusions were drawn from the quadratic regression model and variance analysis, as shown in Figure 7.

Figure 7
Interaction effects diagram: (a) AC interaction response surface plots; (b) AD interaction response surface plots; (c) BC interaction response surface plots; (d) BD interaction response surface plots; (e) CD interaction response surface plots.

The AC interaction (P = 0.0482) was statistically significant. When all other factors were fixed at zero, the signal-to-noise (S/N) ratio was higher with factor A at a high level and factor C at a low level. This suggests that the synergistic effect of higher temperature (which enhances fluidity) and lower flow rate (which ensures precise control) results in more uniform material deposition and stable cooling, ultimately improving planarity.

The AD interaction (P = 0.0388) was also significant. With all other factors set to zero, the S/N ratio was higher when factor A was at a high level and factor D at a low level. The mechanism indicates that a higher nozzle temperature combined with fewer exterior wall layers optimizes material flow, reduces cooling stresses, and minimizes error accumulation, thus enhancing the surface flatness.

The BC interaction (P = 0.0440) showed a significant effect, with higher S/N values observed when both factors B and C were at lower levels. This suggests that at lower lower external perimeters speeds, the extruder has more time to precisely feed the material, minimizing fluctuations in flow rate due to extrusion inertia. Furthermore, the molten material has more time to spread naturally under nozzle pressure, filling inter-layer gaps and producing a more uniform surface.

The BD interaction (P = 0.0099) was highly significant, with higher S/N values when both factors B and D were at low levels. This implies that fewer exterior wall layers allow the infill structure to support surface flatness, while lower external perimeters speeds facilitate a more seamless bond between the infill and exterior walls, preventing wall collapse.

The CD interaction (P = 0.0265) was also significant, with higher S/N ratios when both factors C and D were at low levels. This suggests that a smaller extrusion flow rate combined with fewer exterior wall layers optimizes stress distribution. The reduced flow rate minimizes single-layer errors, while fewer exterior wall layers prevent error accumulation, collectively improving surface flatness.

4.4. Regression analysis

The Design-Expert software was employed to treat the insignificant terms in the model as residuals. Subsequently, regression analysis was performed to derive the regression equation of the model, which is expressed as follows:

(2) $$\mathrm{y}=328–1.61A–0.48B–220.3C+12.3D+0.98AC–0.02AD+0.30BC+9.06\text{X}{10}^{–3}BD–13.44CD+1.87\text{X}{10}^{–3}{A}^2+5.2\text{X}{10}^{–4}{B}^2+0.5D^2$$

4.5. Target optimization

The optimal solution to the system of equations is obtained using MATLAB, with the fmincon function employed for nonlinear constrained optimization. The objective function, denoted as F, is formulated such that fmincon minimizes the negative value of the function Y. Four design variables—denoted A, B, C and D—are introduced, and the feasible domain is defined accordingly. The objective function and design constraints are presented in Equation 3.

(3) $$\begin{array}{l}\mathrm{F}=\min \left(–y\left(A,B,C,D\right)\right)\\ \text{19}0\le A\le \text{23}0;\text{6}0\le B\le \text{2}00;0.\text{96}\le C\le \text{1}.0\text{4};\text{2}\le D\le \text{6}\end{array}$$

The initial values for the iteration are set as X0 = [210, 130, 1.0, 4], and the results of the iteration are presented in Figure 8.

Figure 8
Matlab iteration number and objective function value.

After multiple iterations, the optimal result obtained is -42.427. This optimization leads to a minimized objective function (F), a maximized Y value, and a signal-to-noise ratio of 42.247, with the corresponding parameters being A = 230, B = 60, C = 0.96, and D = 2.

4.6. Experimental verification

Under the optimized conditions, a total of three models were reprinted, with each model measured five times. The maximum flatness error was 0.0136 mm, and the minimum was 0.0080 mm. The signal-to-noise ratio (SNR) values for each model were calculated as 38.4, 37.9, 39.5, respectively. It is evident that the SNR values for the flatness errors of all three models are higher than those obtained from the CCD experiments, indicating that the optimization results are reliable.

The discrepancy between the validation experiment results and the optimization results can be attributed primarily to the fact that the number of layers of the outer wall is constrained to integer values, rather than being continuous. This constraint led to a gap between the final values used in the CCD experimental design and the ideal values that should have been selected, thereby affecting the accuracy of the regression model.

5. CONCLUSION

• ¹⁾

  Based on a review of the literature and studies on roughness and dimensional errors, eight parameters, including nozzle temperature and layer thickness, were selected for experimental screening. From these, four key factors were identified as significantly influencing flatness accuracy: nozzle temperature, external perimeter speed, extrusion flow rate ratio, and wall layer Count.

• ²⁾

  Based on the results from the screening experiments, a Central Composite Design (CCD) experiment was developed to ensure the orthogonality and rotatability of the experimental layout. The star arm value was set at 1.72. The findings revealed that nozzle temperature (A), external perimeter speed (B), extrusion flow ratio (C), and wall layer count (D) significantly impacted the flatness error. Moreover, interactions between factors, specifically AC, AD, BC, BD, and CD, also had a notable effect on the flatness error. A regression analysis was subsequently conducted on the experimental data, which led to the development of a regression model.

• ³⁾

  Based on the regression function, nonlinear optimization was performed using MATLAB, yielding optimal values of A = 230, B = 60, C = 0.96, and D = 2. The function value was maximized, and the corresponding signal-to-noise ratio (SNR) reached 42.247. Under the optimized conditions, three sets of models were generated, and a total of 15 measurements were taken for experimental validation. Although the SNR of the flatness error did not meet the target optimization value, it was significantly higher than the results obtained from 30 model sets in the CCD experiments, indicating that the optimization results are highly reliable.

6. ACKNOWLEDGMENTS

This work is supported by the Key Research Project of Natural Science in Universities in Anhui under Grant No. 2022AH051911.

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Publication Dates

• Publication in this collection
  08 Dec 2025
• Date of issue
  2025

History

• Received
  31 July 2025
• Accepted
  28 Oct 2025