Improvement of the Geometric Accuracy for Microstructures by Projection Stereolithography Additive Manufacturing

Projection stereolithography creates 3D structures by projecting patterns onto the surface of a photosensitive material layer by layer. Benefiting from high efficiency and resolution, projection stereolithography 3D printing has been widely used to fabricate microstructures. To improve the geometric accuracy of projection stereolithography 3D printing for microstructures, a compensation method based on structure optimization is proposed according to mathematical analysis and simulation tests. The performance of the proposed compensation method is verified both by the simulation and the 3D printing experiments. The results indicate that the proposed compensation method is able to significantly improve the shape accuracy and reduce the error of the feature size. The proposed compensation method is also proved to improve the dimension accuracy by 21.7%, 16.5% and 19.6% for the circular, square and triangular bosses respectively. While the improvements on the dimension accuracy by 16%, 17.6% and 13.8% for the circular, square and triangular holes are achieved with the proposed compensation method. This work is expected to provide a method to improve the geometric accuracy for 3D printing microstructures by projection stereolithography.


Introduction
By accumulating materials layer by layer, additive manufacturing (also known as 3D printing) has the ability to create complex 3D structures with high efficiency [1][2][3]. Projection stereolithography, as an important 3D printing approach, has attracted great attention and has been applied to various research fields, including metamaterial, biomedicine, microfluidics, actuators and electronics [4][5][6][7]. In 2005, Sun et al. [8] reported a high-resolution projection stereolithography process by using a digital micromirror device as a dynamic mask. In their study, the performance of projection stereolithography on fabricating 3D microstructures was illustrated and the 3D printing process was mathematically characterized. In the following years, much progress has been made to push the projection stereolithography to become a mainstream 3D printing technology for fabricating microfeatures. Chen et al. [9] presented a bottom-up projection system for the projection stereolithography process and a two-way linear motion approach has been developed for the quick spreading of liquid resin into uniform thin layers. The fabrication speed of a few seconds per layer has been achieved by their proposed system. Tumbleston et al. [10][11][12] proposed the digital light synthesis (DLS) (previously known as continuous liquid interface production or CLIP) process for projection stereolithography, which dramatically improves the printing speed and the surface quality of the printed model. Benefiting from the DLS technique, devices can be 3D printed in minutes with high quality. Shao et al. [13] fabricated a customized optical lens in about two minutes via a DLS-based projection stereolithography 3D printing process while the surface roughness was 13.7 nm. Recently, a volumetric projection stereolithography 3D printing method has been proposed [14][15][16][17], which prints entire complex objects through one complete revolution, circumventing the need for layering. This method was claimed to be particularly useful for high-viscosity photopolymers and multi-material fabrication. Meanwhile, studies have also focused on the system and material of projection stereolithography 3D printing. For example, Huang et al. [18] built a bottom-up projection stereolithography system with a robot arm and realized conform 3D printing through freeform transformation of layers. Chen et al. [19] presented an electrically assisted projection stereolithography system and Shao et al. [20,21] showed a magnetic material suitable for highly accurate 3D printing.
Previous works have made great progress on the process, system and material of projection stereolithography, while the geometric accuracy lacks sufficient study. Using the grayscale control method, Chen et al. [22] successfully improved the surface accuracy of a printed lens and reached a roughness of 6 nm. Zhou et al. [23] developed a geometric calibration system that can calibrate the position, shape, size, and orientation of a pixel and an energy calibration system that can calibrate the light intensity of a pixel. Based on the two systems, the shape accuracy and the uniformity of the power intensity distribution were improved. Researchers have also tried to improve the 3D printing accuracy of projection stereolithography by optimizing the processes [24,25], structures [26,27] and the photosensitive materials [28,29]. The geometric accuracy on the section shape and the size of the projection stereolithography 3D printing plays an important role for the microstructures and the functions of the 3D printed models for various applications, especially in bioengineering [30], optics [31] and microfluidics [32]. For example, the section shape of a 3D printed micro part has a remarkable effect on fog collection performance [33]. Also, accuracy in shape and size is critical to the function of the 3D printed microneedles [34,35].
However, few studies have investigated the 3D printed accuracy of the section shape and size directly. In this paper, the geometric error of the section shape and the dimension was mathematically analyzed and the projection power intensity distribution was studied with simulation tests. Then, a compensation method based on the structure optimization was proposed according to the simulation results. Also, the performance of the proposed compensation method was investigated by the simulation with the representative projection patterns. Finally, by the 3D printing experiments, the proposed compensation method was proved to be effective for improving the geometric accuracy of the projection stereolithography 3D printing. The results in this study are expected to be widely used to fabricate 3D structures for high precision applications.

Geometric Error Analysis
To analyze the geometric error, a projection stereolithography 3D printing system (Figure 1a) was built with a DLP projector that possesses a resolution of 3840 × 2160 pixels and a 405 nm light source (Fuzhou Gyinda Photoelectric Technology Co., Ltd., Fuzhou, China). The pixel resolution of the system was scale to 5 × 5 µm 2 using an UV lens (Universe Kogaku, New York, America, Inc.). The image of each layer is projected onto the surface of the photosensitive resin and the polymerization process is triggered by the UV light. Since the pattern UV light is composed of pixels, the power intensity distribution of the pixel directly affects the geometric accuracy of the cured layer. Equation (1) describes the power intensity distribution of a single pixel [36], which conforms to the Gaussian distribution in Figure 1b. (1) where E( ) is the power intensity at given position , is the peak intensity, and is the Gaussian radius. Generally, the size of the single pixel (L) in projection stereolithography 3D printing system is slightly greater than that of the Gaussian radius. Thus, the light source of a pixel has a distinct influence on the power intensity distribution of adjacent pixels. Meanwhile, the rectangular pixel is not conformal to the shape of the power intensity distribution, which can be seen from the qualitative simulation result in Figure 1b. Both the crosstalk between pixels and the inconsistent shapes between the pixel and its power intensity distribution could cause the geometric errors.
By projecting a square pattern with different dimensions (ranging from 1 pixel to 32 pixels) onto the focus plane, the power intensities of the single pixel and multiple pixels are measured. From the results shown in Figure 1c, the power intensity rises with the increase of the number of pixels while the power intensity becomes saturated when the length of the square region reaches 8 pixels for the built system. Numerically, the intensity distribution of a region containing multiple pixels is obtained from the summation of the Gaussian distribution of each individual pixel. Thus, the evolution of the intensity profile with increasing pixels can be illustrated in Figure 1d. The "flat-top" shape of the intensity profile is obtained when the total size of pixels is much larger than the Gaussian radius. While the power intensity for the "flat-top" represented with can be known by measuring the overall intensity of the projected large square pattern (see Figure 1c) using a dimension much greater than the typical Gaussian radius. Sun et al. [8] gave the general expression of the power intensity at the surface of resin by the Equations (2) and (3).
where E( , ) is the power intensity at given position ( , ) and ( ) is the error function.
As with the case of a single pixel, Figure 1d shows that the power intensity distribution area is always larger than the actual projection area. Moreover, Equation (2) also proves that the effect on the power intensity distribution of the adjacent pixel is nonuniform, depending on the projection pattern. Therefore, due to the unexpected power intensity of the pixels around the projection pattern, the geometric error will be produced between the cured part and the designed model.

Method
To improve the geometric accuracy of projection stereolithography 3D printing, a compensation method based on structure optimization is proposed in this paper. Three representative section features (circle, square and triangle) are included to verify the effectiveness of the proposed compensation method. The designed structure was optimized according to the simulation result of the power intensity distribution. By 3D printing experiments, the geometric errors between the designed structure and its corresponding 3D printed model were demonstrated. Also, the optimized structure was shown to be available to reduce the geometric errors by characterizing with scanning electron microscope (SEM). The photosensitive resin used in this work is composed of polyethylene glycol diacrylate (PEGDA, CAS: 26570-48-9) with a concentration of 99.45 wt%, ethyl (2,4,6-trimethylbenzoyl) phenylphosphinate (TPO-L, CAS: 84434-11-7) by 0.5 wt% as the photoinitiator, and a photoabsorber of Sudan I (CAS: 842-07-9) by 0.05 wt%. By 3D printing and measuring the thickness of suspend beams (Figure 1a), the curing depth curve of the PEGDA resin can be obtained, which is shown in Figure 2.

Numerical Simulation
The simulation experiments were performed to investigate the power intensity distribution with Ansys 2020 software. To simplify the simulation model, a flat plane was divided into pixel cells while each cell was set as a light receiver with a size of 5 × 5 µm 2 . The power intensity distribution on the surface of each cell is defined by Equation (1), in which the Gaussian radius is = 6 µm. The power intensity distribution on the surface of the plane is defined by Equation (2). In the simulation study, we assumed that the LED level is 30%, thus the maximum power intensity of a single pixel was 0.124 mw/mm 2 and the saturated value was 0.564 mw/mm 2 according to Figure 3c. Also, we assumed that the threshold power for photopolymerization was 0.37 mJ/mm 2 while the exposure time was 1s. Then, the threshold power intensity could be calculated, which was 0.37 mw/mm 2 . The power intensity distributions of the circular, square and regular triangular patterns were respectively calculated by the simulation process. As shown in Figure 3a, the power intensity distribution of the circle pattern conforms to the projection contour while the area of the threshold for the photopolymerization is remarkably larger than the area of the designed circle pattern, which leads to the dimensional error between the designed structure and its 3D printed model. The dimensional error for the circle pattern can be described by the Equation (4).
where represents the dimension error for the circle pattern, K is the distance between the center of the circle and the threshold boundary, and R is the radius for the circle pattern.
For the square pattern in Figure 3b, the power intensity on the surface of the resin shows a barrel distribution with a saturated region that exceeds the projection area, indicating the positive geometric error for this case. The outline of the threshold power intensity composed of the curves AB, BC, CD and DA, can be approximately described by the conical section. The curve AB, which can be illustrated by the Equation (5), is selected for the following analysis. = ( − ) (5) where is the distance from the interaction point between the threshold curve and the coordinate axis to the origin point O. is the parameter that determines the curvature, which can be calculated by giving the coordinates of two asymmetrical points on the curve. Based on Equation (5), the geometric error between the threshold outline curve and the projection square pattern can be calculated through Equation (6).
where represent the geometric error for the square pattern along the y + direction. is the length of the square pattern. Due to the symmetry of the projection pattern and the power intensity distribution, the errors for the other three sides of the square pattern are the same as that for the AB side.
From Figure 3c, the power intensity for the triangle pattern presents an elliptic distribution on the surface of the resin and the area with the saturated power intensity is larger than that of the projection pattern. Thus, there will be contour and dimension errors for the 3D printed triangle structure. The geometric error for the MN side can be described as Equation (7).
where is the geometric error for the triangle pattern, is the distance from the line MN to the centroid point O, is the minor axis of the ellipse while is the major axis. Also, since the symmetric pattern and the power intensity distribution, the geometric errors for the other two sides (MF and NF) can be obtained by the same analysis process.
To better understand the effect of the projection pattern on the power intensity distribution, the projection patterns for 3D printing the hole features with different shapes are simulated. Due to the cumulative effect of the power intensity between pixels, there are unexpected pixels with high power intensity inside the hole features, which can be known from the simulation results in Figure 4, leading to the over-cured 3D printing process. Thus, negative dimension errors can be seen from the simulation results compared to the designed hole features. The power intensity distributions for the circular, square and triangular holes are respectively consistent to those of circle, square and triangle projection patterns. Thus, the geometric errors for the hole features could also be calculated according to the Equations (4), (6) and (7).  The simulation results show that the actual power intensity distribution is not exactly consistent with the projection pattern because of the crosstalk between pixels, which brings the geometric errors to the projection stereolithography 3D printing. To improve the geometric accuracy, a compensation strategy based on structure optimization is proposed. For the designed structures with circular, square and triangle sections, the negative compensation strategy for structure optimization is presented due to the positive geometric errors from the numerical simulation tests. For example, the compensation strategy for the square pattern can be illustrated by the Equation (8).
where ′ is the optimized length of the square pattern. While the geometric error is revealed by Equation (5) with the conical section, the optimized structure for the square pattern, which is demonstrated in Figure 5b, can be determined by combining Equations (5), (6) and (8). Likewise, the compensation strategy for the circle pattern is known with the equation of ′ = − ( ′ is the optimized radius) and Equation (4). Figure 5a shows the optimized result for the circle pattern. The compensation strategy for the triangle pattern is obtained by the equation of ′ = − ( ′ is the optimized distance) and Equation (7), and the optimized structure is shown in Figure 5c  To validate the effectiveness of the proposed compensation strategy, the power intensity distributions of the optimized structures were investigated with the simulation tests. The result for the circle pattern shown in Figure 6a indicates that the power intensity distribution of the optimized structure conforms to the designed pattern more precisely compared with the designed structure in Figure 3a. Meanwhile, the power intensity distributions of the optimized square and triangle patterns indicate high 3D printing geometric accuracy, which can be known from Figure 3b,c. Also, comparing with the results in Figure 4, the results of Figure 6d,e show that a higher geometric precision for the circular, square and triangular holes can be reached by the optimized patterns. Thus, the simulation results verify that the proposed compensation strategy is able to dramatically improve the geometric accuracy of the projection stereolithography 3D printing.

Experiments of 3D Printed Parts
To further verify the performance of the proposed compensation method, 3D printing experiments were implemented based on the built system. The thickness of 20 µm, the exposure time of 1 s and the power intensity by 0.5 mw/mm 2 were adopted as the key printing parameters for the experiments in this section. The dimensions for the designed and optimized patterns were the same as those in the simulation tests. The 3D printing results for the circle, square and triangle patterns are listed in Figure 7. The 3D printing result of the designed circle pattern shows a satisfactory shape accuracy with a diameter of 128 µm, which is larger than the designed dimension. From Figure 7b,c, significant geometric errors on the contour shape and the dimension can be found for the square and the triangle patterns. However, by optimizing the structures with the proposed compensation method, a high 3D printing geometric accuracy can be achieved, which is supported by the results in Figure 7d-f. The diameter of the 3D printed structure for the optimized circle pattern is 107 µm, which is close to the designed dimension (100 µm), indicating a high dimension accuracy. By projecting the optimized structure for the square pattern, the contour of the 3D printed model is much closer to square comparing to the result in Figure  7b. The dimension accuracy of the optimized square pattern has been improved, which can be seen from the Figure 7e wherein the diameter of the 3D printed model is 103 µm. Also, the 3D printed contour of the triangle pattern has been corrected to the triangular shape with a high precision dimension by optimizing the projection pattern. The comparison between the original designed and the optimized structures on the dimension accuracy was executed. Batches of models for the circle, square and triangle bosses were 3D printed with the original designed and the optimized structures respectively. The dimensions (refer to Figure 7) for six randomly selected samples of each pattern were measured. Figure 8a shows the measured diameter of each sample 3D printed with the circle pattern while the standard diameter is designed as 100 µm. The results illustrate that the samples printed by the original designed structure led to positive dimension errors and the diameters of samples with optimized structures were close to the designed standard dimension. For the square and triangle patterns (Figure 8b,c), the dimensions of the samples printed with the original designed structures were much larger than the designed standard dimensions (100 µm for the length of square and 86.6 µm for the altitude of the triangle pattern), while the printed dimensions were decreased by optimizing the structures. Furthermore, the mean dimension error for the circle pattern with the original designed structure is 23.3% and the mean error is reduced to 1.6% by using the optimized circle structure. For the square pattern, the mean error of the samples printed with the original structure is 21.3% while that with the optimized structure is 4.8%. Similarly, the mean dimension error for the triangle samples printed with the original structure is 22.7%. Using the optimized structure, 3.1% of the mean error is achieved. Thus, the dimension accuracy of projection stereolithography 3D printing could be improved by 21.7%, 16.5% and 19.6% for the circle, square and triangle structures respectively with the proposed compensation method. For the projection patterns of the hole features, the 3D printing results are shown in Figure 9. By optimizing the pattern for the circular hole according to Figure 5, the 3D printed hole reaches a diameter that meets the design dimension with high precision. For the square hole, the original design pattern leads to the distortion with the small hole size. While the optimized pattern shows the hole feature with the accuracy square shape and the designed size, which can be seen in Figure 9e. Furthermore, the original design pattern for the triangular hole results in an elliptic feature and the smaller size comparing to the design dimension, which is consistent with the simulation result. From the result shown in Figure 9f, it is clear that the 3D printed structure with the optimized pattern possesses an accurate triangular shape with the correct dimension.
The comparison results between the original designed and the optimized structures on the dimension accuracy are demonstrated in Figure 10. Batches of models for the circular, square and triangular holes were 3D printed with the original designed and the optimized structures respectively (refer to Figure 9). The diameter of the designed standard circle is 100 µm, which is the same as the length of the designed square and the length of the designed regular triangle. The negative dimension errors caused by the original designed structures can be detected from the samples for the circular, square and triangular holes, which are shown in Figure 10a-c. However, the samples fabricated with the optimized structure showed the accurate dimensions for each hole feature. Also, the mean dimension errors of samples with different hole features are illustrated in Figure 10d. Specifically, the mean error of the circular hole samples printed with the original designed structure is 16.7% and 0.7% for that of the samples with the optimized circular hole structure. The mean errors before and after the optimization are 18.8% and 1.2% for the samples of the square hole. For the triangular hole, the samples with the original designed structure led to the mean error by 17.1% and this was reduced to 3.3% by using the optimized structure. The results indicate that the proposed compensation method could reduce the dimension error by 16%, 17.6% and 13.8% for 3D printed circular, square and triangular holes respectively.  To further verify the performance of the proposed compensation method, 3D printing experiments with large 3D printing dimension scale were implemented. As shown in Figure 11a,d, two different structures with various features were designed. With the proposed compensation method, the designed structures were regenerated by the transformation of the shape and dimension, which can be seen in Figure 11b,e. The optimized structures were 3D printed (Figure 11c,f) and the models show clear features that are greatly consistent with the designed structures.
The 3D printing experiments showed the same results as the simulation tests. While all of the results prove that the proposed compensation method could significantly improve the geometric accuracy of the projection stereolithography 3D printing.

Conclusions
In this work, the geometric errors of projection stereolithography were mathematically analyzed. Then the simulation tests were conducted to investigate the geometric errors of the projection stereolithography 3D printing with the representative patterns. A compensation method based on the structure optimization was proposed according to the simulation results. By positively or negatively compensating the projection patterns, the geometric errors relative to the design structure could be remarkably decreased. The simulation results and the 3D printing results indicate that the proposed compensation method is able to significantly improve the shape accuracy. The proposed compensation method was also shown to improve the dimension accuracy by 21.7%, 16.5% and 19.6% for the circular, square and triangular bosses respectively. While improvements to the dimension accuracy by 16%, 17.6% and 13.8% for the circular, square and triangular holes could be achieved with the proposed compensation method. The performance of the proposed compensation method was also verified at a large 3D printing dimension scale. This work is expected to provide a method to improve the geometric accuracy for 3D printing microstructures by the projection stereolithography.

Conflicts of Interest:
The authors declare no conflict of interest.