Utilizing Fractals for Modeling and 3D Printing of Porous Structures

Porous structures exhibiting randomly sized and distributed pores are required in biomedical applications (producing implants), materials science (developing cermet-based materials with desired properties), engineering applications (objects having controlled mass and energy transfer properties), and smart agriculture (devices for soilless cultivation). In most cases, a scaffold-based method is used to design porous structures. This approach fails to produce randomly sized and distributed pores, which is a pressing need as far as the aforementioned application areas are concerned. Thus, more effective porous structure design methods are required. This article presents how to utilize fractal geometry to model porous structures and then print them using 3D printing technology. A mathematical procedure was developed to create stochastic point clouds using the affine maps of a predefined Iterative Function Systems (IFS)-based fractal. In addition, a method is developed to modify a given IFS fractal-generated point cloud. The modification process controls the self-similarity levels of the fractal and ultimately results in a model of porous structure exhibiting randomly sized and distributed pores. The model can be transformed into a 3D Computer-Aided Design (CAD) model using voxel-based modeling or other means for digitization and 3D printing. The efficacy of the proposed method is demonstrated by transforming the Sierpinski Carpet (an IFS-based fractal) into 3D-printed porous structures with randomly sized and distributed pores. Other IFS-based fractals than the Sierpinski Carpet can be used to model and fabricate porous structures effectively. This issue remains open for further research.


Introduction
This study relates three issues, namely fractal geometry, porous structure, and threedimensional (3D) printing. These issues and their interplay are described as follows.
First, consider the issue of fractal geometry [1,2]. Fractal geometry provides a new outlook into geometric modeling. It can realistically model objects found in the natural world and living organisms, which is perhaps beyond the scope of Euclidean geometry. Moreover, it can model micro-nano-level details of artificially created objects. Two key concepts of fractal geometry are self-affine shape (popularly referred to as self-similar shape) and fractal dimension. The concept of self-similar shape has been used in biology to model structures, identify patterns, investigate theoretical problems, and measure complexity [3,4]. It has been used in telecommunication to design antennas and relevant devices, ensuring simultaneous multi-frequency transmission [5][6][7]. (Without the fractal antenna, mobile phones do not work.) It has been used in image processing and data compression [8,9], as well as in chemical systems and fracture mechanics studies [10,11]. It has also been used in architectural design to understand the intrinsic complexity and manufacturing of porous structure, fastens fractal geometry, porous structure, and 3D printing. The explanation is as follows. In most cases, porous structures to be fabricated by 3D printing are designed using scaffold-based approaches [31,[66][67][68]. These structures exhibit regularly distributed pores, which is not the case in real porous structures. For example, consider the cases shown in Figures 1 and 2. In Figure 1, a scaffold-based porous structure is shown [31]. Particularly, Figure 1a shows the unit cell which is used to create the scaffold (Figure 1b). Figure 1c shows the 3D printed scaffold (porous structure), and Figure 1d shows a magnified view of the printed porous structure. These kinds of structures do not mimic the real porous structures. Fractal Fract. 2021, 5, 40 3 of 19 Thus, "design for additive manufacturing" of porous structure is an important issue of research and implementation [60][61][62]. Developing methods, enabling design for additive manufacturing of porous structure, fastens fractal geometry, porous structure, and 3D printing. The explanation is as follows. In most cases, porous structures to be fabricated by 3D printing are designed using scaffold-based approaches [31,[66][67][68]. These structures exhibit regularly distributed pores, which is not the case in real porous structures. For example, consider the cases shown in Figures 1 and 2. In Figure 1, a scaffoldbased porous structure is shown [31]. Particularly, Figure 1a shows the unit cell which is used to create the scaffold (Figure 1b). Figure 1c shows the 3D printed scaffold (porous structure), and Figure 1d shows a magnified view of the printed porous structure. These kinds of structures do not mimic the real porous structures. For example, consider the two real porous structures shown in Figure 2. The structure shown in Figure 2a is a clay pot used to grow indoor plants. The other structure shown in Figure 2b is a ceramics-based coffee filter. In both cases (Figure 2), the pores have random sizes and are distributed randomly, as well. This kind of structure exhibits percolation [69]. As a result, fluids and gases can pass through the structure as required. Stochastic point cloud-based methods can create a porous structure like the real ones [61,62,70]. Some simple geometric entities can fill the spaces offered by the randomly distributed points inside a boundary (e.g., polyhedrons [62] and cylindrical tentacles [70]), leading to a realistic porous structure. For example, consider the two real porous structures shown in Figure 2. The structure shown in Figure 2a is a clay pot used to grow indoor plants. The other structure shown in Figure 2b is a ceramics-based coffee filter. In both cases (Figure 2), the pores have random sizes and are distributed randomly, as well. This kind of structure exhibits percolation [69]. As a result, fluids and gases can pass through the structure as required. Stochastic point cloud-based methods can create a porous structure like the real ones [61,62,70]. Some simple geometric entities can fill the spaces offered by the randomly distributed points inside a boundary (e.g., polyhedrons [62] and cylindrical tentacles [70]), leading to a realistic porous structure.
In stochastic point cloud-based porous structuring, the critical issue is how to generate the point cloud itself [61,62]. Depending on the density and distribution of points, the structure changes a lot. For example, consider the case shown in Figure 3. The structures shown in Figure 3 were created using the system developed by Ullah et al. [62]. Figure 3a presents the porous structure. Figure 3b shows the representative cross-sections of the respective porous structures. Figure 3c shows the cross-sections of the respective porous structures when the density of points was increased by eight times. Thus, depending on the density and distribution of points, the underlying porous structure exhibits a different porosity and distribution of pores.  In stochastic point cloud-based porous structuring, the critical issue is how to generate the point cloud itself [61,62]. Depending on the density and distribution of points, the structure changes a lot. For example, consider the case shown in Figure 3. The structures shown in Figure 3 were created using the system developed by Ullah et al. [62]. Figure 3a presents the porous structure. Figure 3b shows the representative cross-sections of the respective porous structures. Figure 3c shows the cross-sections of the respective porous structures when the density of points was increased by eight times. Thus, depending on the density and distribution of points, the underlying porous structure exhibits a different porosity and distribution of pores.
. Point cloud-based porous structuring: (a) porous structures having different boundaries; (b) cross-sections of the ve porous structures; (c) cross-sections when the density of points increased by eight times. Reprinted with perfrom ref. [62].Copyright 2021 ELSEVIER LICENSE.   In stochastic point cloud-based porous structuring, the critical issue is how to generate the point cloud itself [61,62]. Depending on the density and distribution of points, the structure changes a lot. For example, consider the case shown in Figure 3. The structures shown in Figure 3 were created using the system developed by Ullah et al. [62]. Figure 3a presents the porous structure. Figure 3b shows the representative cross-sections of the respective porous structures. Figure 3c shows the cross-sections of the respective porous structures when the density of points was increased by eight times. Thus, depending on the density and distribution of points, the underlying porous structure exhibits a different porosity and distribution of pores.   As far as the stochastic point cloud-based porous structuring is concerned, fractal geometry can play an active role. Particularly, point clouds generated by iterative function system (IFS)-based fractals can help design realistic porous structures. This aspect of fractal geometry (IFS fractal generated point cloud) has not yet been explored from the perspective of porous structuring. This article fills this gap and provides some insights into fractal geometry-based realistic porous structure design and fabrication.
The rest of this article is organized as follows. Section 2 presents the settings to define an IFS fractal for generating stochastic point clouds. Section 3 presents a mathemati-Fractal Fract. 2021, 5, 40 5 of 19 cal procedure to control the self-similarity levels in an IFS-fractal-generated point cloud. Section 4 presents some results of porous structuring using the modified point clouds relevant to an IFS fractal called the Sierpinski Carpet. (The Sierpinski Carpet is described in Sections 3 and 4.) The outcomes of Section 3 are used in Section 4. Section 5 presents the solid CAD models and 3D-printed porous structures based on the outcomes of Section 4. The contents presented in Section 5 demonstrate the efficacy of utilizing fractals in the modeling and 3D printing of realistic porous structures. Section 6 concludes this study.

Settings of IFS-Based Fractals
This section presents the mathematical settings to generate IFS-based fractals. IFS-based fractals are created by some strictly contracting affine maps, as described in [71][72][73][74]. Predefined probabilities control the frequency of participation of each map. To be more specific, we consider the following mathematical settings.
Let {(x i , y i ) | i = 1, . . . , N} be the set of the recursively created point clouds where the initial point is (x 0 , y 0 ) = (0, 0). The relationship between two consecutive points (x i−1 , y i−1 ) and (x i , y i ), i = 1, . . . , N, is as follows: The vector of parameters (a j , b j , c j , d j , e j , f j ) defines the j-th map. The values of the elements of this vector must be set in a way so that they collectively ensure a strictly contracting map as described in [71][72][73][74]. In the mapping processes, multiple maps can be used. Let M be the number of maps to be used. While mapping (x i−1 , y i−1 ) into (x i , y i ), a map out of M maps can be selected randomly. The (random) selection process must be controlled. As such, probabilities p j ∈ [0, 1], ∀j ∈ {1, . . . , M} are assigned to the maps so that p 1 + . . . + p M = 1. Thus, after performing the mapping process N times, the frequency of the j-th map participating in the mapping process must be about p j × N. To achieve this, first, the cumulative probability of each map, denoted as cp j , is calculated as follows: The two consecutive cumulative probabilities produce an interval. This interval, in turn, determines the weight (w j ) of the respective map. For the first and last maps, the weights are w 1 = [0, cp 1 ) and w M = [cp M−1 , cp M ], respectively. For others, the weights are w j = [cp j−1 , cp j ), ∀j∈{2, . . . , M − 1}. This means that the weights are mutually exclusive intervals that partition the interval [0, 1] so that {w j | j = 1, . . . , M} = [0, 1] and w j ∩ w j + 1 = Ø, ∀j∈{1, . . . , M − 1}. Thus, the following relationship holds: If a random number denoted as r i ∀ [0, 1] (i = 1, 2, . . . ) belongs to w j , the corresponding map participates in the mapping process, others not. This ensures the relative frequencies of the respective maps according to p j provided that the number of iterations N is relatively large.
Based on the above consideration, the IFS-based point cloud creation process follows an algorithmic approach. This approach can be defined by an algorithm denoted as the IFS Algorithm. The main processes of the IFS Algorithm are presented in Algorithm 1. The point cloud denoted as PC produced by the IFS Algorithm is the stochastic point cloud representing the given fractal shape. For example, Figure 4 shows nine point-clouds representing nine different fractal shapes (tree, seahorse, fern leaf, snowflakes, and the Sierpinski Carpet). The settings of the affine maps {(a j , b j , c j , d j , e j , f j , p j ) | j = 1, . . . , M} can be found in [71][72][73][74]. The Sierpinski Carpet (Figure 4i) is considered in this study to model a porous structure. As far as manufacturing is concerned, the IFS Algorithm generated point cloud must undergo some transformation, controlling the self-similarity levels. For example, consider the case in Figure 5. Figure 5a shows a point cloud of fern leaf (same as Figure 4c). Figure 5b shows the same shape; this time, the self-similarity is controlled up to level three. The self-similarity level controlling mechanism employs a set of one-to-one maps. Sometimes, selected affine maps out of M maps can be omitted for the sake of controlling the level of self-similarity. See [72,73] for the details. It is worth mentioning that self-similar structures created by tree-like fractals have been used to reinforce a 3D printed structure [75,76]. In this case, solid models, not the point cloud, of the self-similar segments are used, which is not the focus in this study. However, the manufacturability of these self-similar segments limits the level up to which the self-similar tentacles can be fabricated with required accuracy [72,73,77]. to level three. The self-similarity level controlling mechanism employs a set of one-to-one maps. Sometimes, selected affine maps out of M maps can be omitted for the sake of controlling the level of self-similarity. See [72,73] for the details. It is worth mentioning that self-similar structures created by tree-like fractals have been used to reinforce a 3D printed structure [75,76]. In this case, solid models, not the point cloud, of the self-similar segments are used, which is not the focus in this study. However, the manufacturability of these self-similar segments limits the level up to which the self-similar tentacles can be fabricated with required accuracy [72,73,77].  to level three. The self-similarity level controlling mechanism employs a set of one-to-one maps. Sometimes, selected affine maps out of M maps can be omitted for the sake of controlling the level of self-similarity. See [72,73] for the details. It is worth mentioning that self-similar structures created by tree-like fractals have been used to reinforce a 3D printed structure [75,76]. In this case, solid models, not the point cloud, of the self-similar segments are used, which is not the focus in this study. However, the manufacturability of these self-similar segments limits the level up to which the self-similar tentacles can be fabricated with required accuracy [72,73,77].

Controlling Self-Similarity Levels
This section presents how to control self-similarity levels of IFS fractal-driven point clouds to create models of porous structures.
A process consisting of five steps is proposed to control the self-similarity levels of IFS fractal-driven point clouds. The description of these steps is as follows.
Step 1: This step generates a seed point cloud denoted as PCS = {(x si , y si )|i = 0, 1, . . . , N} by an IFS-based fractal or by any other means. The number of points PCS must be chosen carefully so that the desired pore size can be achieved after performing the modifications described in Steps 2, . . . , 5.
Step 2: This step generates the first generation modified point cloud representing the first level of the underlying self-similar fractal or porous structure denoted as The following one-to-one map is applied to each point in PCS to get its counterpart in PC G−1 . Thus, the following relationship holds: The mapping parameters (a c , b c , c c , d c , e c , f c ) are denoted as critical parameters. The goal here is to bring the PCS to a region resulting in the first level self-similar shape of the underlying fractal or porous structure.
Step 3: This step generates the second generation modified point cloud denoted as . . , N} representing the second level of the underlying self-similar fractal or porous structure. The following one-to-one map is applied to each point in PC G−1 to get its counterpart in PC G−2 . As defined in (4), one of the affine maps of the underlying IFS fractal out of M maps are randomly selected to map a point ( Thus, the random selection process is defined in Algorithm 2, which is like "Map" segment of IFS Algorithm 1: Step 4: This step generates the third, fourth, . . . , generations of modified point clouds, denoted as PC G−3 , PC G−4 , . . . , respectively, repeating the mapping process defined in Step 3. The repetition process can be terminated based on the preference of the modeler. For example, if the modeler wants the third level self-similar shapes, Step 4 will be repeated one time only. Similarly, if the modeler wants the fourth level self-similar shapes, Step 4 will be repeated twice. Step 5: This step first generates a filler point could denoted as PCF = {(x F,k , y F,k ) | k = 0, . . . , L} and fills the empty regions PC G−1 , PC G−2 , PC G−3 , . . . , following the mapping processes defined in Step 2,3, . . . , respectively. Let PCF be the point clouds generated in this step. of the modeler. For example, a modeler can apply Step 1, Step 2, Step 5, Step 3, Step 4, Step 4, and Step 4 successively. Another modeler can apply Step 1, Step 2, Step 3, Step 4, Step 5, successively. This means that depending on the need or preferences of a model, the steps and their sequence can be determined. There is no restriction as such.

Modeling Porous Structures Using Sierpinski Carpet
This section shows how to modify the Sierpinski Carpet-driven point cloud (an IFS fractal, shown in Figure 4i) so that it becomes a porous structure model. The steps presented in the previous section will be used to achieve this goal.
First, consider the settings of the IFS fractal called the Sierpinski Carpet. The settings of the Sierpinski Carpet [74] are listed in Table 1. As listed in Table 1, the Sierpinski Carpet needs eight affine maps with equal weights (probability) of 1/8. The maps create a point cloud that resides in a square-shaped boundary given by vertices (0, 0) (1, 0), (1, 1), and (0, 1). The distribution of points creates self-similar squares, as shown in Figure 6. As seen in Figure 6, the self-similarity is scaled by 1/3. The maps symmetrically distribute the points around the scaled squares. As seen in Figure 6, the self-similar levels above the third level are not visible. If more points are injected (i.e., N is increased), the higher levels might be visible. This issue is, however, beyond the scope of this study.
and Step 4 successively. Another modeler can apply Step 1, Step 2, Step 3, S successively. This means that depending on the need or preferences of a mo and their sequence can be determined. There is no restriction as such.

Modeling Porous Structures Using Sierpinski Carpet
This section shows how to modify the Sierpinski Carpet-driven point fractal, shown in Figure 4i) so that it becomes a porous structure model. T sented in the previous section will be used to achieve this goal.
First, consider the settings of the IFS fractal called the Sierpinski Carpe of the Sierpinski Carpet [74] are listed in Table 1. As listed in Table 1, the Sier needs eight affine maps with equal weights (probability) of 1/8. The maps cloud that resides in a square-shaped boundary given by vertices (0, 0) (1, (0, 1). The distribution of points creates self-similar squares, as shown in Fig in Figure 6, the self-similarity is scaled by 1/3. The maps symmetrically points around the scaled squares. As seen in Figure 6, the self-similar lev third level are not visible. If more points are injected (i.e., N is increased), the might be visible. This issue is, however, beyond the scope of this study.  Let us apply Steps 1, …,5 and modify the point cloud in Figure 6. Let us apply Steps 1, . . . ,5 and modify the point cloud in Figure 6.
Step 2 is applied as follows: the first-level self-similar shape is created by scaling the points of PCS by a factor of 1/3, resulting in PC G−1 = {(x G−1,i , y G−1,i ) | i = 0, 1, . . . , N}. For this, the following formulation is used: As such, the values of the critical parameters are a c = 1/3, b c = 0, c c = 0, d c = 1/3, e c = 1/3, and f c = 1/3. The point clouds PCS and PC G−1 are shown in Figure 7b.
Step 3 is applied as follows. The affine maps shown in Table 1 are used to create the second-generation modified point cloud PC G−2 from PC G−1 , as defined in Algorithm 2. The point clouds PCS, PC G−1 , and PC G−2 are shown in Figure 7c.
Step  Figure 7f. Note that starting from the fourth level, the self-similar squares are not clearly manifested. This is because of the shortage of points in the seed, i.e., the PCS. For example, Figure 8 shows a case of the point clouds up to the fourth level. This time, a smaller number of points in the seed point cloud is used compared to the case shown in Figure 7. As seen in Figure 8, starting from the third level, the self-similar squares were not clearly manifested. Whether or not this is desirable from the context of porous structuring needs further investigation. This issue is kept out of the scope of this study, however.
Step 5 is applied as follows. Recall that Step 5 is an additional step, which can be applied before or after Steps 3 and 4. The goal is to fill the voids in certain regions of a given level of self-similarity. In this case, a filler point cloud PCF = {(x F,k , y F,k ) | k = 0, 1,

Solid Modeling and 3D Printing
This section shows some solid models of porous structures, as well as structures. Figure 10 shows the point clouds controlled up to the fourth level and i The point clouds used in this case are shown in Figure 10a. The solid CAD in Figure 10b, is created using commercially available software that can c

Solid Modeling and 3D Printing
This section shows some solid models of porous structures, as well as the 3D printed structures. Figure 10 shows the point clouds controlled up to the fourth level and its solid model. The point clouds used in this case are shown in Figure 10a. The solid CAD model, shown in Figure 10b, is created using commercially available software that can convert a raster graphics entity into a vector graphics entity using voxel-based geometric modeling. Figure 11 shows another similar case. This time, the point clouds, shown in Figure 11a, controlled up to the fifth level is considered. The corresponding solid CAD model is shown in Figure 11b. This way, a user can apply the procedure described in Sections 3 and 4 to digitize a porous structure model given the level-controlled point clouds of the IFS fractal.

Solid Modeling and 3D Printing
This section shows some solid models of porous structures, as well as the 3D printed structures. Figure 10 shows the point clouds controlled up to the fourth level and its solid model. The point clouds used in this case are shown in Figure 10a. The solid CAD model, shown in Figure 10b, is created using commercially available software that can convert a raster graphics entity into a vector graphics entity using voxel-based geometric modeling. Figure 11 shows another similar case. This time, the point clouds, shown in Figure 11a, controlled up to the fifth level is considered. The corresponding solid CAD model is shown in Figure 11b. This way, a user can apply the procedure described in Sections 3 and 4 to digitize a porous structure model given the level-controlled point clouds of the IFS fractal.  To obtain more insights into the fabrication of porous structures, a commercially available 3D printer is used. Several 3D-printed structures have been produced. Three of the results are shown in this section. Figure 12 shows the point clouds controlled up to level five, the corresponding 3D-printed structure, and a magnified view of the structure. As seen in Figure 12, the porous structure exhibits a complex network among the pores in the structure. This is partly because of the limitation of the solid CAD model and partly because of the limitation of the 3D printer.  To obtain more insights into the fabrication of porous structures, a commercially available 3D printer is used. Several 3D-printed structures have been produced. Three of the results are shown in this section. Figure 12 shows the point clouds controlled up to level five, the corresponding 3D-printed structure, and a magnified view of the structure. As seen in Figure 12, the porous structure exhibits a complex network among the pores in the structure. This is partly because of the limitation of the solid CAD model and partly because of the limitation of the 3D printer. To obtain more insights into the fabrication of porous structures, a commercially available 3D printer is used. Several 3D-printed structures have been produced. Three of the results are shown in this section. Figure 12 shows the point clouds controlled up to level five, the corresponding 3D-printed structure, and a magnified view of the structure. As seen in Figure 12, the porous structure exhibits a complex network among the pores in the structure. This is partly because of the limitation of the solid CAD model and partly because of the limitation of the 3D printer.
the results are shown in this section. Figure 12 shows the point clouds controlled up level five, the corresponding 3D-printed structure, and a magnified view of the structu As seen in Figure 12, the porous structure exhibits a complex network among the pores the structure. This is partly because of the limitation of the solid CAD model and par because of the limitation of the 3D printer.  Figure 13 shows the point clouds controlled up to level five, the corresponding 3Dprinted structure, and a magnified view of the structure. This time, the filling operation was carried out, as descried in the previous section. Thus, in the point clouds (Figure 13a), the first and second level self-similar square regions are not filled with points. The corresponding porous structure (Figure 13b,c) exhibits a complex network among pores. As seen in Figure 13, the porous structure exhibits a complex network among the pores in the structure. Like the previous case, this is partly because of the limitation of the solid CAD model and partly because of the limitation of the 3D printer.   (Figure 13b,c) exhibits a complex network among pores. seen in Figure 13, the porous structure exhibits a complex network among the pores in t structure. Like the previous case, this is partly because of the limitation of the solid CA model and partly because of the limitation of the 3D printer. What if the structure is scaled down while printing it? When the structure is p duced by reducing the scale, the pore network becomes more complex. As a result, ra domness in the pore sizes increases. This is because of the limitation of the 3D printer. F example, consider the case shown in Figure 14. The point clouds shown in Figure 13 a used to print the structure shown in Figure 14. The structure is scaled down by half duri the printing, resulting in a more complex network among the pores than that of Figure What if the structure is scaled down while printing it? When the structure is produced by reducing the scale, the pore network becomes more complex. As a result, randomness in the pore sizes increases. This is because of the limitation of the 3D printer. For example, consider the case shown in Figure 14. The point clouds shown in Figure 13 are used to print the structure shown in Figure 14. The structure is scaled down by half during the printing, resulting in a more complex network among the pores than that of Figure 13.
What if the structure is scaled down while printing it? When the structure is produced by reducing the scale, the pore network becomes more complex. As a result, randomness in the pore sizes increases. This is because of the limitation of the 3D printer. For example, consider the case shown in Figure 14. The point clouds shown in Figure 13 are used to print the structure shown in Figure 14. The structure is scaled down by half during the printing, resulting in a more complex network among the pores than that of Figure 13. In addition to using a voxel-based solid modeling of porous structure from levelcontrolled point clouds of IFS fractals, other solid modeling approaches can be used. In this respect, one of the obvious modeling approaches is to model each point of the point clouds by using special polyhedrons (e.g., prismoid). For example, consider the case shown in Figure 15. In this case, the modeling concept is to model each point of an IFS-generated point cloud (in this case, a point cloud representing Sierpinski Carpet created by the IFS Algorithm defined in Algorithm 1) by a hexagonal cylinder (see Figure 15a). This way, multiple point clouds can be created and solid modeled. The point clouds look alike but are not the same due to the stochasticity involved in point cloud creation process (see IFS Algorithm 1). In Figure 15b, three point clouds are shown to be modeled by the proposed concept. The solid models of these point clouds are shown in Figure 15c. Finally, these solid models can be combined to print a porous structure. One printed porous structure is shown in Figure 15d. In addition to using a voxel-based solid modeling of porous structure from levelcontrolled point clouds of IFS fractals, other solid modeling approaches can be used. In this respect, one of the obvious modeling approaches is to model each point of the point clouds by using special polyhedrons (e.g., prismoid). For example, consider the case shown in Figure 15. In this case, the modeling concept is to model each point of an IFSgenerated point cloud (in this case, a point cloud representing Sierpinski Carpet created by the IFS Algorithm defined in Algorithm 1) by a hexagonal cylinder (see Figure 15a). This way, multiple point clouds can be created and solid modeled. The point clouds look alike but are not the same due to the stochasticity involved in point cloud creation process (see IFS Algorithm 1). In Figure 15b, three point clouds are shown to be modeled by the proposed concept. The solid models of these point clouds are shown in Figure 15c. Finally, these solid models can be combined to print a porous structure. One printed porous structure is shown in Figure 15d.
In synopsis, IFS-generated stochastic point clouds provide a flexible means to model porous structures, and thereby, to fabricate them using 3D printing or additive manufacturing.

Concluding Remarks
There are many application areas where porous structures with randomly sized and distributed pores are required, e.g., biomedical applications (producing implants), materials science (developing cermet-based materials with desired properties), engineering applications (objects having controlled mass and energy transfer properties), and smart ag-  In synopsis, IFS-generated stochastic point clouds provide a flexible means to model porous structures, and thereby, to fabricate them using 3D printing or additive manufacturing.

Concluding Remarks
There are many application areas where porous structures with randomly sized and distributed pores are required, e.g., biomedical applications (producing implants), materials science (developing cermet-based materials with desired properties), engineering applications (objects having controlled mass and energy transfer properties), and smart agriculture (devices for soilless cultivation). In most cases, a scaffold-based method is used to design porous structures. This approach fails to produce randomly sized and distributed pores. Thus, more effective porous structure design methods are required. In this respect, methods capable of filling the spaces offered by a stochastic point cloud are quite effective. IFS-based fractals manifest stochastic point clouds. Thus, these point clouds are candidates for porous structure design. This possibility is explored in this study, and some promising results are obtained.
A mathematical procedure is developed to create stochastic point clouds using the affine maps of a predefined IFS-based fractal. In addition, a method is developed to modify a given IFS fractal-generated point cloud. The modification process controls the self-similarity levels of the fractal and ultimately results in a model of porous structure exhibiting randomly sized and distributed pores. The model can be transformed into a 3D CAD model using voxel-based modeling or other means for digitization and 3D printing.
The efficacy of the proposed method is demonstrated by transforming the Sierpinski Carpet (an IFS-based fractal) into 3D printed porous structures. The condition of achieving fractal geometry-created porous structure must follow a definite sequence of mapping combining some predefined one-to-one and affine mappings. The sequence of mapping must control the self-similar structures as well as fill some preselected regions. The sequence of mapping for the Sierpinski Carpet-based porous structuring is elucidated. The proposed mathematical procedure is effective in this regard, as demonstrated in this study using the case of Sierpinski Carpet-generated point clouds. Other IFS-based fractals than the Sierpinski Carpet can be used to model and fabricate porous structures effectively. This issue remains open for further research.
The four different printed porous structures presented here demonstrate that the fractal-based porous structuring creates a complex network among randomly sized and distributed pores in the structure.
Fractal geometry is well known for modeling and quantifying complex shapes observed in the natural world, living organisms, and artificial objects (particularly in micronano scale). In this respect, the concepts of self-similarity and fractal dimensions are extensively used. On the other hand, fractal geometry's ability to produce stochastic points has not yet been explored. As demonstrated in this study, IFS-based fractals are an effective means to produce a stochastic point cloud, which can model realistic porous structures. Thus, this study extends the scope of fractal geometry to a large extent.
Conducting experimental studies employing real porous structures (including cermetbased porous structures), is expensive and requires a long time. Before performing experimental studies employing real porous structures, preliminary experiments can be performed using 3D printed artificial porous structures. The results of preliminary experiments can be used to optimize the experimental studies employing real porous structures. Thus, the outcomes of this study can be used to reduce the expense and time of porous structure-related experimental studies. This issue is also open for further investigation.