Research on intra-layer partition adaptive slicing algorithm for fused deposition modeling Open Access

Additive Manufacturing (AM), commonly known as 3D printing, is a rapidly expanding advanced manufacturing technology (Shanmugam et al., 2021). It has extensively used in high-tech industries, offering innovative solutions for modern production challenges (Jabbari and Abrinia, 2018; Bermudo et al., 2023). Unlike traditional subtractive manufacturing, which removes material from a larger block, AM selectively adds material only where needed, enabling the creation of complex structures layer by layer from digital models. As a transformative production method, AM provides significant advantages, including faster production from design to final part, the capability to fabricate intricate and complex geometries, minimal setup time, substantially reduced material waste and the ability to maintain the same tools and fixtures without requiring changes to molds, even when designs are modified (Park et al., 2022; Anerao et al., 2024). These advantages have made AM an indispensable tool in industries such as aerospace, healthcare, automotive and consumer goods, where its ability to produce customized, lightweight, geometrically sophisticated and small-batch components is highly valued (Vinodh, 2023; Savran et al., 2023; Tshephe et al., 2022; Zhao et al., 2023).

AM comprises various techniques, including Stereolithography (SLA), selective laser melting and fused deposition modeling (FDM), each serving distinct material and application needs (Rasheed et al., 2023). Among these, FDM has emerged as the most widely adopted method due to its unique process characteristics. (Wang et al., 2019; Doshi et al., 2022; Vyavahare et al., 2020a, 2020b; Panjwani et al., 2020). Compared to other AM techniques, FDM features a simple equipment structure and ease of operation. It operates by extruding thermoplastic materials through a heated nozzle, depositing them layer by layer to create three-dimensional structures without the need for molds or extensive post-processing. This process allows for internal structures to be sparsely filled rather than completely solid, which not only minimizes material waste but also reduces printing time, enabling the efficient fabrication of complex geometries. Moreover, this process demonstrates versatility by supporting multicolor printing and compatibility with various thermoplastic materials such as PLA, ABS and PETG, making it adaptable to a wide range of manufacturing needs. These characteristics give FDM a wide range of applications, making it particularly well-suited for visual aids, prototypes, conceptual designs and craft models (Dey and Yodo, 2019). Its ability to support multicolor printing and offer features like customization, cost-effectiveness, visual appeal and faster production from design to final part makes FDM indispensable and difficult to replace in these fields.

However, even though the process advantages of FDM are already outstanding compared to other manufacturing techniques, there remains significant room for improvement within the technology itself. Many scholars are continuously exploring and researching FDM’s forming efficiency, surface quality and mechanical properties to further enhance its performance (Chacón et al., 2021; Vyavahare et al., 2020a, 2020b; Tamașag et al., 2023; Buj-Corral et al., 2021). Each process optimization for FDM is of great significance for further enhancing its effectiveness in targeted applications and advancing its role in accelerating industrial innovation and development. Therefore, this paper takes the FDM process as the research focus and optimizes its slicing algorithm to further balance the relationship between printing efficiency and surface finish.

The principle of forming in FDM determines that layer thickness is an important process parameter affecting both the surface finish (Wankhede et al., 2020; Lalegani and Lalegani Dezaki, 2021; Pérez et al., 2018; Vyavahare et al., 2020a, 2020b) of the model and forming efficiency (LI et al., 2024). For models with the same inclination angle, a larger layer thickness results in higher forming efficiency but poorer surface finish; conversely, a smaller layer thickness leads to lower forming efficiency but better surface finish. To address this issue, scholars have conducted extensive research starting from the layer thickness in slicing, developing various slicing algorithms.

Based on the uniform layer thickness slicing algorithm (Xu et al., 2018), the adaptive slicing algorithm (Dolenc and Mäkelä, 1994; Mao et al., 2019; Rosa and Graziosi, 2019) was first proposed, which uses different layer thicknesses for different layer height ranges of the model. For layer height ranges where the model surface has a certain inclination relative to the forming direction, a smaller layer thickness is used. To achieve adaptive layer thickness, scholars have proposed various methods. Sabourin et al. (1996) presents a stepwise refinement method, which first applies uniform layer thickness using the maximum layer thickness for the model, then calculates the residual height between layers step by step. Each thick layer is further divided as needed to ultimately achieve the desired surface finish. Hu et al. (2022) proposes an adaptive 3D printing slicing method based on precise apex height and improved aspect ratio. Unlike the method that directly compares area differences, this method measures printing errors by calculating the union and intersection of the profile across different slicing planes, using the difference between the union and intersection to address specific models, such as cylindrical shapes with slopes. Liu et al. (2020) proposes a method for extracting outer contour lines through angle grouping. Based on the extracted contour lines, adaptive slicing is performed by slicing the model with a series of planes perpendicular to the z-axis. The degree of variation in the intersecting contour lines obtained from the slicing is used to mathematically compute the adaptive layer thickness for each layer of the 3D model. The introduction and application of the adaptive slicing algorithm enhance the balance point between forming efficiency and surface finish compared to the uniform thickness slicing algorithm.

To further balance the relationship between surface finish and forming efficiency of the model, scholars have proposed an intra-layer partitioning algorithm based on regional division (Wang et al., 2021), built upon adaptive slicing. This method divides different regions within the same layer and applies different layer thicknesses according to each region. Based on different partitioning criteria, the methods can be mainly divided into two categories: local adaptive slicing methods (Tyberg and Helge Bøhn, 1998) and adaptive slicing with internal and external partitioning (Sabourin et al., 1997; Wang and Wang, 2022). The former mainly addresses situations in complex models where different surface finish requirements exist within the same layer thickness region. For example, in the case of an asymmetrical object with a vertical surface on the left and an inclined surface on the right, the vertical side can be formed with a larger layer thickness, while the inclined side requires a smaller layer thickness to improve surface finish. The latter focuses on models where the exterior needs to be smooth and flat, while the interior is not affected by the visual impact of layer stacking. Therefore, the model is divided into internal and external regions: the internal region uses larger layer thicknesses to improve forming efficiency, while the external region uses smaller layer thicknesses to enhance surface finish.

The above content reviews the literature on balancing surface finish and forming efficiency with respect to the process parameter of layer thickness, focusing primarily on improvements to adaptive slicing algorithms and innovations in intra-layer partitioning algorithms. Most studies on intra-layer partition adaptive slicing algorithms concentrate on theoretical research and software implementation, lacking practical forming tests or only conducting tests on specific models.

This paper proposes a novel intra-layer partition adaptive slicing algorithm, which features algorithmic innovations in handling layer objects, dividing intra-layer partition regions and defining path regions compared to existing region-based adaptive slicing algorithms. The algorithm uses Boolean operations on the sectional contours of layer objects as the basis for adaptive slicing. Based on the adaptive slicing results, it selectively performs intra-layer partitioning, effectively avoiding the negative impact on forming efficiency caused by small cross-sectional layer objects or large layer thicknesses layer objects that are unsuitable for intra-layer partitioning. To address the integration challenges between regions with different layer thicknesses, the algorithm introduces “transition zones,” which are intermediate regions designed to ensure smooth and seamless connections between internal and external regions. These transition zones are established during path region planning, enabling effective bonding and uniform material deposition. Unlike algorithms in previous studies that are tailored to specific models, the proposed algorithm is versatile and applicable to models of various shapes, sizes and geometric features. This algorithm enhances efficiency while maintaining surface finish and minimizing the impact on the bonding strength of the printed model. To validate the feasibility and reliability of the algorithm, a variety of experiments were conducted. A laser confocal microscope was used to measure the surface roughness of the specimens, providing an assessment of their surface finish. The specimens were longitudinally sectioned to observe the forming quality of the intra-layer partitions. Additionally, the forming time of the proposed method was compared with that of traditional slicing algorithms, and models with different features were tested to evaluate its contribution to forming efficiency and validate its generalization capability. Tensile tests were also performed to analyze the algorithm’s impact on bonding strength. These experiments demonstrated the algorithm’s improved performance in multiple aspects, highlighting its potential for enhanced efficiency and reliability.

The intra-layer partition adaptive slicing algorithm is a novel algorithm improved upon the traditional adaptive slicing algorithm, enabling models to achieve higher printing efficiency while maintaining the surface finish provided by the conventional adaptive slicing algorithm. This algorithm is of significant importance to vendors and individual users who provide printing services for large-sized FDM models, as it can greatly enhance their production efficiency.

An STL model is a geometric entity composed of a series of triangular facets, characterized by the principle that adjacent triangular facets adhere to point-to-point and edge-to-edge relationships. The triangular facets within the model are stored in an unordered manner, and there is no logical relationship between the facets. STL files have two storage formats: one is a text format that saves coordinates and offers good readability, while the other is a binary format that is more space-efficient. This paper reads STL files saved in text format. As shown in Figure 1, it displays the coordinate information of the geometric model stored in the STL file. The text is divided into several sections, with each section representing a facet. The first line of a facet is led by the keyword “facet normal,” followed by numbers that represent the coordinates of the normal vector of the current facet. The third to fifth lines are led by the “vertex” keyword, with subsequent numbers representing the coordinates of the three vertices of the current triangular facet.

Based on the characteristics of the text format STL file, the information of the triangular facets is read and uniformly stored in a triangular facet list called “triangle,” facilitating subsequent intersection operations between the triangular facets and the slicing planes.

The creation of layer objects is a key step in achieving FDM (Zhao and Guo, 2020). The principle of FDM involves converting the material into a molten state and laying it down from a one-dimensional point to form a two-dimensional surface, which is then layered to create the entire entity. The creation of each layer object impacts the final printing result.

The establishment of layer objects in the intra-layer partition adaptive slicing algorithm consists of two parts: initially coarsely slicing with a larger layer thickness and then refining based on that larger layer thickness. The first part involves an initial slicing with a layer thickness of 0.2 mm, creating the initial layer object. This initial layer object includes the layer height and the intersection profiles formed by the intersection of the current layer’s slicing plane with the model’s triangular mesh, resulting in the intersection contour lines. The second part involves refining the layer thickness to 0.1 mm, which is half of the initial layer thickness, for models with significant changes in cross-sectional contour. This refinement ensures a smooth surface finish for the model. The choice of 0.2 mm as the layer thickness is based on its common use as a layer thickness parameter for a 0.4 mm nozzle diameter. Furthermore, when subdividing to a layer thickness of 0.1 mm, it allows for a consistent filament output and good forming results without the need to change the nozzle. The 0.2 mm thickness is subdivided by half to avoid inter-layer gaps during intra-layer partitioning. The following sections will provide a detailed introduction to both parts.

The creation of the initial layer objects consists of three parts: obtaining unordered intersection line segments from the intersection of the slicing plane and the model’s triangular facets, stitching the intersection line segments to form a closed contour and adjusting the direction of the contour.

An intersection plane layer height sequence is established with an initial layer thickness of 0.2 mm. The minimum layer height is below the lowest point of the model, while the maximum layer height is above the highest point of the model, as shown in Figure 2. Since the triangular facets are stored in an unordered manner, traversing each triangular facet with the slicing plane for each layer height is inefficient. This paper adopts a scanning plane method for intersecting with the triangular facets.

The scanning plane, like the slicing plane, is a plane parallel to the XY plane, with its height varying along the z-axis according to the layer height sequence from low to high. The difference is that the scanning plane dynamically updates the list of facets based on the height changes; this list contains all the triangular facets in the model that intersect with the scanning plane.

When using the scanning plane method to intersect with the model, all triangular facets of the model are first sorted in ascending order based on their lowest z-coordinate. This ordering of triangular facets can reduce the number of traversals when dynamically updating the scanning facet list, eliminating the need to traverse all triangular facets each time. During the continuous scanning process of the scanning plane from the lowest point of the model to the highest point, the scanning facet list dynamically adds and removes triangular facets, ensuring that the list always contains only those facets that can intersect with the scanning plane. Triangular facets that are below the scanning plane’s lowest point or above its highest point are removed from the list. At each layer height, all facets in the STL model that can intersect with the scanning plane are traversed to perform the intersection calculations.

The result of the intersection is a series of unordered intersection line segments obtained at different layer heights. To facilitate the subsequent establishment of the layer object contour, these segments need to be stitched together, connecting line segments with the same vertex coordinates to form closed polygons with geometric significance.

The intersection results for a specific layer height obtained through the scanning plane method are shown in Figure 3. Due to the unordered nature of the triangular mesh, the resulting intersection line segments are also stored in an unordered manner. Two line segments that are connected end-to-end in spatial position may not be adjacent in storage, and independent segments cannot directly describe the geometric information of the current layer’s profile. Therefore, it is necessary to stitch the intersection line segments together to form a closed polygon with geometric significance. The key to stitching the intersection line segments is to identify the spatial adjacency relationships between the segments, stitching together segments that share a common point. However, sequentially selecting and traversing segments for stitching can be inefficient. In this paper, a dictionary lookup method is used for stitching the segments.

Refer to the book “Computer-Aided Manufacturing Practice – Python Implementation of 3D Printing Path Planning” by Dr Lin Zhiwei from Zhejiang University. A data structure called LinkPoint has been established using the dictionary lookup method, which simultaneously contains attributes for points and line segments, as shown in Figure 4. Compared to the data structure for points, it includes an additional attribute called “other.” In contrast to the data structure for line segments, it omits one point object. This data structure allows for the quick retrieval of the other endpoint when one endpoint of the segment is known, thereby preserving the inherent relationship between the two points of the segment. A connection between two coincident but distinct points is established using the dictionary type in Python. The key of the dictionary is the shared coordinates of the two points, while the value is a list that stores the two point objects. In practical operation, the endpoints of all line segments are stored using the new data structure LinkPoint (hereinafter referred to as LP). The LP is then added to a point list, where the X and Y coordinates of the LP serve as the keys. Points with the same coordinates will be stored in the point list corresponding to the same key. The data structure LinkPoint establishes the relationship between the two points on the same line segment, while the key-value pair data structure establishes the relationship between different points at the same coordinate. By leveraging this data structure and Python’s built-in dictionary class for fast query functionality, the information of neighboring points can be determined more efficiently. This approach offers significantly improved speed compared to traditional direct traversal methods for segment stitching.

After stitching all the unordered line segments, the model contours for each layer are obtained and stored in the initial layer object.

The model contours stored in the initial layer object may not consist of a single closed polygon; they can also exhibit parallel and inclusion relationships among closed polygons, as shown in Figure 5. To distinguish the inner and outer boundaries of the model and delineate the correct filling area, it is necessary to recognize and adjust the directions of the obtained model contours. The contour direction of the outer boundary is defined as counterclockwise, while the contour direction of the inner boundary is defined as clockwise.

This paper adopts the ray-casting method to identify the boundary type (inner or outer) of the polygon, thereby determining the correct path filling region. As shown in Figure 6, a ray is cast from an arbitrary vertex of the polygon to be identified, and the number of intersection points between the ray and the other polygon contours is calculated. By counting these intersection points, we can determine how many times the ray crosses the polygon contours from the starting point to infinity, thus identifying the boundary type of the current polygon. It is important to note that the number of intersection points does not necessarily equal the number of crossings. In cases where the ray passes exactly through the vertex of the polygon, the determination of whether it constitutes a valid intersection point must be based on the distribution of the segments that include the vertex relative to the ray. When both line segments that include the intersecting vertex are on the same side of the ray, as shown by Point 1 on Ray2 in Figure 6, it is considered an invalid intersection point and does not serve as a basis for determining whether the polygon is an inner or outer boundary. Conversely, when the two line segments are located on opposite sides of the ray, as indicated by Point 2 on Ray3 in Figure 6, it is considered a valid intersection point. If the number of valid intersection points is even, the polygon is classified as an outer boundary and should be oriented counterclockwise; if the number is odd, the polygon is classified as an inner boundary and should be oriented clockwise.

The formation and stitching of the intersection line segments, along with the identification and adjustment of the model contour direction, ensure the accurate and rapid establishment of the initial layer object, providing a data foundation for the implementation of the intra-layer partitioning algorithm.

Layer object refinement is a component of the adaptive slicing strategy and serves as a prerequisite for implementing the intra-layer partitioning algorithm discussed in this paper. Common adaptive slicing strategies include two approaches: (1) initially performing uniform slicing with the minimum layer thickness and then merging layers, and (2) initially performing uniform slicing with the maximum layer thickness and then refining the layers. This paper uses a method that combines uniform thickness with adaptive slicing through a stepwise refinement strategy, where the model is first coarsely sliced using a larger layer thickness and then finely sliced using a smaller layer thickness. The need for refinement is determined by assessing the area difference between the contours of adjacent layers. For layer height ranges of the model that require refinement, layer object refinement is performed.

The specific method uses the projection area method, which performs a Boolean difference operation on the contours of adjacent layers projected onto the XY plane and calculates the area of the resulting region (Wang et al., 2007; Pan et al., 2014). Based on the rate of change of the area, it determines whether refinement is needed for the layers. Unlike traditional refinement slicing algorithms, which can choose from various layer thickness combinations, this paper’s refinement slicing algorithm re-slices the layer intervals with an area difference between adjacent contours at half the initial layer thickness. It uses a loop to traverse the triangular surface and uses direct intersection and direct stitching methods to insert new layer objects. The projection area method can effectively avoid issues that the area method cannot, such as torsional bodies with the same cross-sectional area and models with the same cross-section but different growth directions, as shown in Figure 7.

The use of 1 / 2 layer thickness for slicing is because the refined partition’s layer height range represents a potential intra-layer partition height range. In subsequent operations, the same layer may be divided into internal and external regions. The external region uses the refined smaller layer thickness of 0.1 mm to ensure surface finish, while the internal region uses the initial larger layer thickness of 0.2 mm to improve printing efficiency. When the layer thicknesses of the internal and external regions do not maintain an integer ratio, inter-layer gaps can occur, as shown in Figure 8(a), which can adversely affect bonding strength. If a smaller ratio, such as 1/3 or 1/4, is selected, it may hinder the smooth extrusion process and the quality of the forming effect due to significant differences in layer thickness using the same nozzle. Therefore, this paper adopts a method of taking 1 / 2 of the initial layer thickness to ensure the compatibility of layer thicknesses between the internal and external regions, thus avoiding the generation of inter-layer gaps, as illustrated in Figure 8(b).

The comparison of layer objects before and after refinement is shown in Figure 9. Figure a represents the original model, Figure b shows the slicing effect before the refinement of layer objects and Figure c illustrates the slicing effect after the refinement. It is evident that in the layer intervals where the surface contours change, the layer thickness has been subdivided for better detail.

After the refinement of layer objects, a complete sequence of adaptive layer thickness has been obtained, providing data support for subsequent layer partitioning and path planning.

Intra-layer partitioning is the core component of the algorithm presented in this paper. Its purpose is to divide the contour regions of the same layer into internal and external areas within the layer height range of the adaptive slicing with smaller layer thicknesses. The internal region is formed directly using a larger layer thickness to create the equivalent of two layers of smaller thickness, while the external region maintains the original layer thickness. This approach aims to further enhance printing efficiency while ensuring that the model’s contour is formed with a smaller layer thickness. Intra-layer partitioning allows for the creation of two different layer thicknesses during the path filling of the layer object due to the distinctions between the internal and external regions. For the same path trajectory, the time taken for forming with a larger layer thickness is essentially the same as that with a smaller layer thickness; increasing the layer height does not lead to an increase in formation time. When the external region is sufficiently small within the small layer height range, the intra-layer partitioning strategy can nearly halve the forming time while ensuring surface quality.

Intra-layer partitioning is performed only within the small layer height range. For the large layer height range, a greater layer thickness has already been used, so intra-layer partitioning is unnecessary. In the case of the small layer height range, when the cross-sectional area is too small, performing intra-layer partitioning does not save time compared to direct formation and is therefore not required. Intra-layer partitioning is only conducted when the cross-sectional area is sufficiently large and located further from the end face, with the aim of improving the efficiency of the model formation. The flowchart of intra-layer partitioning is shown in the Figure 10, illustrating the key steps of the process.

Within the partitioned layer height range, the junctions between different layer thicknesses may encounter inter-layer bonding strength issues due to thickness mismatches, where the layer thickness of the internal region is not an integer multiple of that of the external region, as shown in Figure 8(a). In this paper, the layer height is preprocessed during the subdivision of the layer object, such that the initial layer thickness is twice that of the refined layer thickness. For the formation of the internal region, a layer thickness of 0.2 mm is selected, making the thickness of the internal region twice that of the external region. During the formation process, the stacking of the two external layer thicknesses can precisely match the thickness of the internal region, thereby avoiding the generation of longitudinal gaps. A schematic of the intra-layer partitioning is illustrated in Figure 8(b).

For the division of the internal and external regions, this paper uses a method that first calculates the intersections of the projected contours of adjacent cross-sections in the XY plane after initial slicing, followed by a subsequent offsetting process. First, the projected Boolean intersection of the slice contours of adjacent layers is calculated to obtain a new contour, denoted as contour c, which is contained within both contour a and contour b, as shown in Figure 11. This contour represents the original region of the internal area. If contour c is directly used as the internal region for intra-layer partitioning, it would result in the external region corresponding to the sliced layer of contour a being unable to print with a small layer thickness. Therefore, contour c needs to be offset inward, reserving the external areas corresponding to slices a and b. The offset result will be used as the final intra-layer region. Theoretically, a higher area proportion of the internal region within the entire contour c implies a greater proportion of 0.2 mm layer thickness, which can lead to improved forming efficiency. However, considering the filling rules during path filling, a larger proportion of the internal region is not always better for contour c. The specific reasons for this will be elaborated in detail in the next section on path generation.

The intra-layer partitioning leverages the advantages of different layer thicknesses by applying varying thicknesses to distinct regions: a larger layer thickness is used for the internal region to enhance printing efficiency, while a smaller layer thickness is applied to the external region to ensure surface quality. This approach improves printing efficiency without compromising the surface finish of the model.

The process of slicing involves reducing a three-dimensional model to two-dimensional plane contours. This section focuses on generating line fill paths within the contours based on the slicing results. These paths are designed to meet the requirements of the molten deposition molding 3D printing process and are used to create the solid material of the model on each layer. By stacking these layers, the printing of the model is completed.

Contour infill paths and parallel directional infill paths are two commonly used infill strategies. The contour infill path consists of a series of offset contour curves that successively offset inward to cover the entire fill area, as shown in Figure 12(a). This contour path can effectively restore the shape of the model’s contour. However, the curved and complex contour trajectories can lead to frequent acceleration and deceleration of the printer’s motors on each axis, resulting in poor speed consistency and low printing efficiency. The directional parallel fill path consists of a series of evenly spaced parallel line segments, as shown in Figure 12(b). This method fills the entire area using inclined line segments in a specific direction. Due to its continuity in one direction, the parallel fill path effectively reduces the frequent acceleration and deceleration of the printer’s axes and extrusion motors compared to contour paths, minimizing platform vibrations. Therefore, parallel fill paths are typically used for most areas within the interior.

In the path generation process, it is important to consider not only the type of filling path but also the filling density. Contour filling maintains a 100% filling density due to its algorithm, while parallel filling depends on the printing position of the model to determine whether to use sparse or dense filling. The filling density of sparse filling is set according to the process requirements, represented by the spacing between paths. In contrast, dense filling adjusts the spacing based on the width of the nozzle to ensure that the two parallel lines are closely joined. Sparse filling is commonly used in the interior, while dense filling is applied to the end surfaces of the model.

This paper uses a combination of contour paths and parallel paths to fill the sliced contours. The contour filling is used near the outer surface of the contour to ensure surface accuracy. The internal filling method varies depending on whether it is a layer within a partitioned area or a layer close to the end face. The end face and areas near the end face use dense filling, while internal filling for non-end faces and non-layer-partitioned layers uses parallel path filling. This approach efficiently covers the entire area and provides support for the model. For non-end face layer partitioned layers, considering the issue of interlayer bonding strength, contour filling is used to delineate the boundary between the inner and outer regions, as shown in Figure 13(a), to ensure the strength at the junction of the inner and outer regions. If only parallel filling is used, the result will be as shown in Figure 13(b), where the parallel filling of the inner and outer regions cannot achieve sufficient bonding, leading to a reduction in the bonding strength within the model.

Before filling the fill areas, it is essential to define single-connected domains and multi-connected domains. A polygonal region is defined as a single-connected domain if, at every scanning height, it contains only one continuous filling line segment. Conversely, if a polygonal region contains multiple filling line segments at any scanning height, it is classified as a multi-connected domain. The distinction between single-connected and multi-connected domain is relative to the filling direction. When using fixed-direction parallel filling paths to fill the areas, if the filling region is a multi-connected domain, the filling lines will be cut into multiple segments by the polygon. During the actual printing process, this can lead to path disconnections, backtracking or frequent movement of the nozzle, thereby affecting print quality and efficiency. Therefore, for model sections where the filling area is a multi-connected domain, it is necessary to divide the multi-connected domain into single-connected regions to obtain continuous filling curves as much as possible. This paper divides multi-connected regions based on the method of segmenting the multi-connected area using concave and peak points. The multi-connected regions before and after segmentation are shown in Figure 14.

In the layer height range of the intra-layer partition, adjacent layers share the same internal region but have different external regions. The differences in the external regions can lead to varying gaps or overlaps in the contour filling of that area after filling, as shown in Figure 15(a), which affects the bonding strength between the internal and external regions. In this paper, a transition zone is created in the external region, using parallel line filling to bridge the gaps between adjacent layers in the external area, ensuring a strong bond with the same internal region, as illustrated in Figure 15b.

The result of the path filling is stored in the corresponding layer object, thereby obtaining the path trajectories of the entire model at various layer heights.

The export order of G-code determines the printing order. Compared to the traditional export of uniform layer thickness slicing data, the data from the intra-layer partition adaptive slicing algorithm has some differences in the settings of printing order and filament extrusion amount.

In the layer height range of the intra-layer partition, the printing order of the nozzle does not proceed to the next layer after completing one layer; instead, it follows a “inside first, outside second” sequence. First, the internal region is printed with a large layer thickness, and then the external region is printed in two passes from bottom to top using a small layer thickness, as shown in Figure 16(a). The choice of “inside first, outside second” is made because the contour of the formed internal region can serve as the inner boundary for the external region, providing a suitable inner boundary for the directionally parallel filling of the external region. When switching between the internal and external regions, it is important to adjust the extrusion amount. Since this algorithm does not specifically sort multiple single-connected regions derived from the partitioning of multi-connected regions in spatial order, when printing a model with a complex multi-connected region, as shown in Figure 16(b), the nozzle may create a protrusion inside that is higher than the external region after completing the internal region. When switching to a single-connected region of the external area, this could lead to interference between the nozzle and the already formed parts. To avoid this issue, the height of the nozzle should be adjusted after completing a single single-connected region to prevent interference.

Using uniform layer slicing, adaptive slicing and the intra-layer partition adaptive slicing algorithm, various test models are processed and formed using a 3D printer. Experiments are designed to analyze the effects of the intra-layer partition algorithm on the surface finish, forming efficiency and structural strength of the formed parts.

Experiments were conducted by Yadav et al. (2023) to test the surface finish of parts made by FDM. The results indicate that layer thickness has a significant impact on surface finish. Reducing layer thickness can improve the surface finish of the model, and the adaptive slicing algorithm is one of the effective methods to enhance surface finish. The intra-layer partition adaptive slicing algorithm performs intra-layer partitioning based on the adaptive slicing algorithm, theoretically leading to similar surface finishes for the same model. To verify this assumption, a comparative experiment on surface finish was designed. The surface finish of the formed part is related to its corresponding surface roughness measurement values (Temiz, 2024). By measuring the surface roughness of test models formed using different slicing algorithms and comparing the differences in roughness values, the surface finish of the formed parts can be analyzed.

The author developed SLICER software capable of performing uniform layer slicing, adaptive slicing and intra-layer partitioning adaptive slicing. To avoid the influence of printer and material variability on the experimental results, all test specimens in this paper were printed using the same printer and material described below. The printer used is a self-assembled FDM 3D printer, supporting a maximum print size of 200 × 200mm × 180mm and a printing temperature range of 170°C to 300°C, with a nozzle diameter of 0.4 mm, a printing speed set at 1500 mm/min and a travel speed (non-printing movement) set at 3000 mm/min. The material used is white PLA filament with a diameter of 1.75 mm from the same manufacturer. To avoid moisture absorption, the filament was stored in a drying box during use. To prevent potential damage to the surface structure of the test specimen caused by traditional contact measurement devices, which could affect the test results, the roughness testing instrument chosen is the Olympus latest industrial laser confocal microscope OLS4000, as shown in Figure 17. This instrument performs non-contact measurements of the specimen’s surface and provides high-resolution 2D and 3D morphological observations to validate the effectiveness of the algorithm (Liao et al., 2023).

Using different slicing strategies to slice the test model, as shown in Figure 18, the dimensions of the bounding box are 20 × 20 × 22mm. To achieve adaptive slicing, it is necessary to maintain a certain variation in the cross-sectional contour in the forming direction; therefore, side A is designed as a raised arc surface. The slicing parameters are shown in Table 1, and other printing parameters are as follows: support type: no support; filling method: parallel line filling; nozzle temperature: 210°C; build plate temperature: 55°C. The sliced file is exported as a G-code file to a storage card.

Using a 3D printer, four test specimens were produced, as shown in Figures 19(a)–(d), processed by four slicing algorithms: constant thickness of 0.1 mm, constant thickness of 0.2 mm, adaptive slicing and intra-layer partition adaptive slicing.

Using a laser confocal microscope, the surface morphology of the test specimen’s testing surface was observed, and its surface roughness was measured. The side surface of the specimen was used as the testing surface, as shown in Figure 20(a), with the middle upper region selected as the area to be measured. This area belongs to the effective layer height range of the adaptive and intra-layer partitioning adaptive slicing algorithms, as illustrated in Figure 20(b). The surface of the testing area was observed in two dimensions, and the results are shown in Figure 21. From the figures, it can be seen that specimens a and b used uniform layer slicing with layer thicknesses of 0.1  and 0.2 mm, respectively. Specimens c and d used adaptive layer thickness, with a larger layer thickness of 0.2 mm in areas with low curvature change and a smaller layer thickness of 0.1 mm in areas with significant curvature changes to ensure surface finish, consistent with the designed layer thickness. The intra-layer partition adaptive slicing algorithm achieved the effect of adaptive layering. The three-dimensional morphology of the testing area is shown in Figure 22, visually reflecting the variation in layer thickness among different specimens within the same layer height range, further validating the effectiveness of the intra-layer partition adaptive algorithm in adapting layer thickness.

Five points at different locations on the test surface were selected for roughness testing. For each location, three sets of roughness measurements were taken along lines parallel to the forming direction, as shown in Figure 23. The average of the three sets was taken as the line roughness for that location, and the average of the line roughness from the five locations was considered as the roughness of the specimen’s side surface. The corresponding surface roughness test results for each specimen are shown in Table 2. Specimen “a” exhibited a smaller surface roughness due to its smaller layer thickness, while specimen “b” displayed a larger surface roughness due to its larger layer thickness. Specimens “c” and “d”, which used the same adaptive slicing strategy, maintained a consistent roughness level, falling between that of specimens “a” and “b”. In the layer height range where Position 5 is located, the cross-sectional contour changes are minimal and it does not fall within the effective layer height range of the adaptive slicing algorithm, so a larger layer thickness of 0.2 mm is used. Consequently, the surface roughness at Position 5 is significantly higher than at Positions 1, 2 and 3 for both the adaptive slicing algorithm and the intra-layer partition adaptive slicing algorithm.

From the experimental results, it can be concluded that the intra-layer partition adaptive slicing algorithm achieves surface finish comparable to that of the adaptive slicing algorithm. Building on the adaptive slicing approach, the intra-layer partition algorithm does not affect the surface finish of the formed parts.

To verify the actual effect of intra-layer partitioning during the forming process using the intra-layer partition adaptive slicing algorithm, an observation experiment was designed to examine the internal regions of the model.

The model shown in Figure 24(a) was cut along both the build direction and the direction perpendicular to it to observe the internal fill effect. As shown in Figures 24(b) and (c), Figure 24(b) clearly reveals the contour fill for the outer boundary, parallel fill in the transition zone, contour fill between the internal and external regions and the parallel fill within the internal region. Figure 24(c) illustrates the difference between intra-layer partition layer height ranges and uniform layer height ranges in the vertical dimension; the black arrows indicate the uniform slicing layer height ranges, while the yellow arrows indicate the intra-layer partition layer height ranges. In all adaptive slicing ranges requiring intra-layer partitioning, the intended intra-layer partition effect was successfully achieved.

Experimental results indicate that the intra-layer partition adaptive slicing algorithm achieves the effect of intra-layer partitioning, and the forming results are satisfactory.

To verify the generalizability of the intra-layer partition adaptive slicing algorithm and assess its impact on build efficiency, models with different features were printed and their build times were recorded. By comparing the build times of four slicing strategies across different models, the influence of the intra-layer partition adaptive slicing algorithm on FDM build efficiency was analyzed.

The test models are shown in Figure 25 and include a rotational model, a twisted model, an irregular model, a hollow model and a model with a complex surface profile. The specific parameters for each model are detailed in Table 3. Each model was processed and fabricated using the four slicing strategies listed in Table 4, with their respective build times recorded. The improvements in build efficiency achieved by the intra-layer partition adaptive slicing algorithm, as compared to the traditional adaptive slicing algorithm, were then analyzed and quantified.

The models formed using the intra-layer partition adaptive slicing algorithm are shown in Figure 26, with good fabrication quality. The experimental results are presented in Figure 27. Figure 27(a) is a bar chart of build times, where Method 1, 2, 3 and 4 correspond to uniform layer slicing with 0.1  and 0.2 mm layer heights, adaptive slicing and the intra-layer partition adaptive slicing algorithm, respectively. Figure 27(b) shows the efficiency gains of the intra-layer partition adaptive slicing algorithm relative to the adaptive slicing algorithm for different test models. Except for Model 4, the intra-layer partition adaptive slicing algorithm demonstrated positive improvements in build efficiency across models. Models 2, 3, 5, 7, 8 and 9 showed the most significant efficiency gains. These models share a common feature: a relatively large cross-sectional area in the plane perpendicular to the build direction, allowing the larger layer thickness of 0.2 mm to be applied extensively to the internal region within the adaptive layer height range, saving build time. In contrast, Model 10, which has a cross-sectional area similar to Model 9, did not show the same positive efficiency effect despite using the larger layer thickness for internal region filling.

The difference in build efficiency between Models 9 and 10, despite their similar cross-sectional areas, arises from the idle travel of the printhead. The essence of the intra-layer partition adaptive slicing algorithm is to use a larger layer thickness to directly form the internal regions of two layers from traditional slicing within the small layer height range of adaptive slicing. For the same path trajectory, the time taken for forming with a larger layer thickness is essentially the same as that for a smaller layer thickness. Thus, increasing the layer height does not lead to an increase in build time. As a result, a larger cross-sectional area allows for a greater proportion of the 0.2 mm layer thickness, leading to improved build efficiency. In principle, when the external region is sufficiently small, the intra-layer partitioning strategy can save nearly half of the time. During the actual forming process, the nozzle, in addition to the printing path, also travels through a non-printing travel distance when switching between different areas of the model.

In the actual slicing process, taking Model 10 as an example, the slicing results of two adjacent layers at the bottom are compared to evaluate the filling paths generated by the intra-layer partition adaptive slicing algorithm and the traditional adaptive slicing algorithm, as shown in Figure 28. When the internal and external fill densities are the same, the intra-layer partitioning adds a contour path that separates the internal and external areas. In this process, the internal region is formed with a layer thickness of 0.2 mm, while the external region is formed with a layer thickness of 0.1 mm. In contrast, the traditional adaptive slicing algorithm forms the entire section using two layers with a uniform layer thickness of 0.1 mm. This is the intuitive difference between the two approaches. From the perspective of printing time, the intra-layer partition adaptive slicing algorithm essentially substitutes the forming time of the second-layer internal filling path in the traditional adaptive slicing algorithm with the contour filling time that distinguishes internal and external regions. The shorter the contour filling time relative to the internal filling time, the greater the time savings achieved by the intra-layer partitioning, resulting in a more significant improvement in forming efficiency. This deduction has been validated in several models. For example, in Model 4, the small cross-sectional dimensions resulted in the contour filling time for separating internal and external areas after intra-layer partitioning being longer than the normal internal filling time. Consequently, the intra-layer partitioning algorithm not only failed to improve forming efficiency compared to the traditional adaptive slicing algorithm but also led to a decrease in efficiency. In contrast, for Model 10, the situation is different. According to the above reasoning, the contour paths separating the internal and external areas of Model 10 should save more time than the larger internal filling paths in the middle. Further analysis reveals that the nozzle’s path includes not only the printing paths but also the empty travel distances when adjusting the printing area. Compared to the traditional algorithm, the intra-layer partitioning algorithm introduces an additional transition zone. When the transition zone is complex, it needs to be divided into more multi-connected areas. Each division of a single-connected area implies that there will be some empty travel when searching for the next segment, and this empty travel is related to the storage order of the single-connected areas. This paper has not optimized this process algorithmically. It is precisely because the concatenation of empty travels from multiple single-connected areas in the transition zone increases the time for the intra-layer partition algorithm, thereby reducing forming efficiency. This can be demonstrated with Models 9 and 10, which have similar sizes and cross-sectional areas but differing forming efficiencies. The primary reason lies in the complex surface contours of Model 10. As shown in Figure 29, multiple single-connected areas were defined when filling the transition zone; in contrast, Model 9 has a simple outer contour, which eliminates the need to repeatedly search for path starting points when filling the transition zone, allowing the advantages of the intra-layer partition algorithm to be better realized.

Based on the analysis of the experimental results, the intra-layer partition adaptive slicing algorithm demonstrates an improvement in forming efficiency for most models. The best efficiency gains are observed in models with simple cross-sectional shapes, significant curvature changes and larger dimensions in length, width and height. In contrast, large models with complex cross-sections and smaller models do not show significant advantages over traditional adaptive methods in terms of forming efficiency. For large models with complex cross-sections, the improvement in printing efficiency is limited due to the division of intricate transition zones into multiple single-connected regions. This division of regions results in increased idle traveling time for the nozzle between single-connected regions, leading to longer forming times. For models with smaller cross-sections, intra-layer partitioning can sometimes be counterproductive, as the combined forming time for internal regions and transition zones may exceed that of forming two contour layers using traditional methods. To address these issues, future work could focus on optimizing the region sorting algorithm after the division of single-connected areas for large models. By grouping spatially adjacent regions together during storage, the idle traveling time between single-connected regions could be minimized, further enhancing the printing efficiency of models with complex contours. For models with smaller cross-sections, future studies could investigate the minimum cross-sectional area parameter suitable for the intra-layer partitioning algorithm. In this approach, intra-layer partitioning would only be applied when the cross-sectional area exceeds a critical threshold, ensuring both efficiency and effectiveness.

Cuiffo et al. (2017) indicates that the printed filament exhibits higher bonding strength when fused at elevated temperatures, and implementing intra-layer partitioning may affect this bonding strength. To investigate the impact of intra-layer partitioning on the strength at the bonding points of the model, tensile specimens were designed based on the characteristics of the intra-layer partition adaptive slicing algorithm and FDM forming process, as shown in Figure 30. The slicing results of four slicing algorithms were controlled to break at the minimum cross-section, allowing for a comparison of the influence of different slicing algorithms on bonding strength. To prevent fractures at both ends and to enhance the strength of the end faces, the filling density at the ends was increased. A gradually varying cross-sectional diameter was used in the middle to accommodate adaptive layer thickness, with the minimum diameter at the center to avoid stress concentration, and transitions were made using arcs.

Using SOLIDWORKS software, the test specimen was designed and exported as an STL file. The slicing results were obtained using self-developed slicing software, which performed uniform thickness, adaptive, and intra-layer partition adaptive slicing, and the sliced results were exported as G-code files. After printing the specimens with a self-assembled FDM 3D printer, tensile tests were conducted using a CTM8050 microcomputer-controlled electronic universal testing machine. A total of four groups of samples were tested, with three identical specimens in each group, resulting in 12 test specimens in total, as shown in Figure 31. The test results were averaged for the three specimens in each group.

As shown in Figure 32, all tensile test specimens fractured at the minimum cross-section, consistent with the designed fracture location. The corresponding tensile curves for each test specimen are illustrated in Figure 33, and the tensile strengths for each slicing strategy are summarized in Table 5. The average tensile load for the test specimens with a layer thickness of 0.1 mm is greater than that of the specimens with a layer thickness of 0.2 mm, which aligns with the findings reported by Popescu et al. (2018). The average tensile load for the specimens using the adaptive slicing algorithm falls between the two, while the average tensile load for the specimens using the intra-layer partition adaptive slicing algorithm is lower than that of the specimens with a layer thickness of 0.2 mm.

The intra-layer partition adaptive slicing algorithm has a certain degree of impact on structural strength compared to the traditional adaptive slicing algorithm, as the intra-layer partition can reduce the bonding strength in the direction of model formation. However, for applications such as visual aids, prototypes, conceptual designs and craft models, where customization, cost-effectiveness, visual appeal and faster production from design to final part are critical, this slight reduction in strength is not significant and does not affect its usability.

The intra-layer partition adaptive slicing algorithm proposed in this paper improves printing efficiency for most models while maintaining the same surface finish quality as traditional adaptive slicing algorithms. In the tested models, the forming efficiency increased by up to 26.39% compared to traditional adaptive slicing algorithms, with potential for further improvement. Additionally, the algorithm provides good internal forming results. However, it has a certain impact on the bonding strength of the models compared to traditional adaptive slicing algorithms. Nevertheless, in FDM applications that prioritize efficiency and surface finish, such as prototype fabrication, conceptual design verification and craft model production, this drawback does not have a significant effect. These scenarios typically emphasize rapid manufacturing, cost-effectiveness and aesthetic quality over structural strength, making the proposed algorithm well-suited for these use cases.

The proposed intra-layer partition adaptive slicing algorithm demonstrates great potential for improving FDM printing of large-scale models by balancing efficiency and surface finish quality.

To further enhance the forming efficiency of the algorithm, especially for models with complex cross-sectional profiles, spatial positioning sorting of regions after the division of single-connected areas can be incorporated during the path-filling stage to minimize the nozzle’s idle travel distance. Moreover, the algorithm currently performs intra-layer partitioning at layer thicknesses of 0.1  and 0.2 mm. Expanding its application to include more layer thickness combinations, such as 0.1 , 0.2 , 0.3  and 0.4 mm, could improve printing efficiency by matching larger internal region layer thicknesses, thereby optimizing the balance between print quality and efficiency.

The concept of intra-layer partitioning can also be extended to multi-material model printing, where materials are allocated to different intra-layer regions based on specific requirements. This extension has the potential to significantly enhance the forming quality and functionality of printed models. By improving the adaptability and efficiency of FDM, such advancements would further solidify its role in accelerating industrial innovation and development.