Utilizing Fractals for Modeling and 3D Printing of Porous Structures

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Article Utilizing Fractals for Modeling and 3D Printing of Porous Structures

AMM Sharif Ullah 1,* , Doriana Marilena D’Addona 2 , Yusuke Seto 3, Shota Yonehara 3 and Akihiko Kubo 1

1 Division of Mechanical and Electrical Engineering, Kitami Institute of Technology, 165 Koen-cho, Kitami 090-8507, Japan; [email protected] 2 Department of Chemical, Materials and Industrial Production Engineering, University of Naples, Federico II, Piazzale Tecchio 80, I-80125 Naples, Italy; [email protected] 3 Graduate School of Engineering, Kitami Institute of Technology, 165 Koen-cho, Kitami 090-8507, Japan; [email protected] (Y.S.); [email protected] (S.Y.) * Correspondence: [email protected]; Tel.: +81-157-26-9207

Abstract: 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 Citation: Ullah, A.S.; D’Addona, levels of the fractal and ultimately results in a model of porous structure exhibiting randomly sized D.M.; Seto, Y.; Yonehara, S.; Kubo, A. and distributed pores. The model can be transformed into a 3D Computer-Aided Design (CAD) Utilizing Fractals for Modeling and 3D Printing of Porous Structures. model using voxel-based modeling or other means for digitization and 3D printing. The efficacy of Fractal Fract. 2021, 5, 40. https:// the proposed method is demonstrated by transforming the Sierpinski Carpet (an IFS-based fractal) doi.org/10.3390/fractalfract5020040 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. Academic Editor: Shengwen Tang; This issue remains open for further research. Giorgio Pia and E Chen Keywords: fractal geometry; porous structure; 3D printing; geometric modeling; point cloud Received: 5 April 2021 Accepted: 28 April 2021 Published: 30 April 2021 1. Introduction Publisher’s Note: MDPI stays neutral This study relates three issues, namely fractal geometry, porous structure, and three- with regard to jurisdictional claims in dimensional (3D) printing. These issues and their interplay are described as follows. published maps and institutional affil- First, consider the issue of fractal geometry [1,2]. Fractal geometry provides a new iations. 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 Copyright: © 2021 by the authors. shape) and fractal dimension. The concept of self-similar shape has been used in biology Licensee MDPI, Basel, Switzerland. to model structures, identify patterns, investigate theoretical problems, and measure This article is an open access article complexity [3,4]. It has been used in telecommunication to design antennas and relevant distributed under the terms and devices, ensuring simultaneous multi-frequency transmission [5–7]. (Without the fractal conditions of the Creative Commons Attribution (CC BY) license (https:// antenna, mobile phones do not work.) It has been used in image processing and data creativecommons.org/licenses/by/ compression [8,9], as well as in chemical systems and fracture mechanics studies [10,11]. 4.0/). It has also been used in architectural design to understand the intrinsic complexity and

Fractal Fract. 2021, 5, 40. https://doi.org/10.3390/fractalfract5020040 https://www.mdpi.com/journal/fractalfract Fractal Fract. 2021, 5, 40 2 of 19

aesthetics of patterns and structures [12,13]—the list of its applications continues. Like the concept of self-similar shape, the concept of fractal dimension [14–17] has been extensively used to quantify the complexity of objects or phenomena. For example, it has been used in manufacturing engineering to process signals [18], detect tool-wear [19,20], and quantify roughness [21]. Secondly, consider the issue of porous structure. Porous structures perform better than their non-porous counterparts on many occasions. As a result, these structures have been used in many areas, including energy [22–24], biomedical [25–28], structural design [29–31], construction [32–34], and safety [35]. The remarkable thing is that there is an interplay between porous the structure and fractal geometry (particularly, fractal dimension). The description is as follows. Fractal dimension can quantify the intrinsic complexity of porous structures [36–38], including the cermet-based porous structures [39]. It is shown that the properties of cermet-based porous structures such as thermal conductivity, electric resistivity, permeability and diffusion, fluidity, and mechanical strengths depend on their fractal dimensions [40]. Thus, fractal dimension has become a design parameter to optimize cermet-based porous structures’ chemical composition and porosity. To be more specific, consider the articles in [41–44]. Wang et al. [41] found that a certain percent of silica fume in low-heat Portland cement concrete can refine the pores and change its shrinkage behavior. Moreover, the fractal dimension of the porous surface exhibits a linear correlation with the concrete shrinkage. Therefore, fractal dimension can be used as a design parameter for designing cermet-based materials with the right shrinkage behavior. In a similar study, Wang et al. [42] investigated the effectiveness of adding fly ash, polymer-based reinforcement, magnesium oxide, and shrinkage-reducing admixtures in enhancing the frost resistance of concrete. In this study, the fractal dimension played a vital role in establishing the right design rule (right compositions of additives mentioned above). In addition, the frost resistance of concrete can be increased by increasing the fractal dimension by choosing the right combination of fly ash and polymer-based fiber reinforcements [43]. Moreover, fractal dimension has a more profound effect on the abrasion resistance of concrete than the porosity and helps determine the right composition of silica fume and fly ash, ensuring cracking and abrasion resistance properties [44]. Consider the other issue, i.e., 3D printing or additive manufacturing. 3D printing can revolutionize the way in which products have been designed and manufactured [45]. Nowadays, it is used to produce nutritious food (complex protein) [46], tablets for visually impaired patients [47], and wooden composites extracting shape information from micro X-ray computed tomography [48]. It is used to produce prototypes of culturally significant complex objects [49], complex structures of hydroponics (soilless cultivation) [50], hierarchically porous micro-lattice electrodes for lithium-ion batteries [51], and even weft-knitted flexible textile [52,53]. The list continues. Advanced research studies have been carried out to make 3D printing even more robust and cost-effective, resulting in novel devices [54–56], geometric modeling techniques [57,58], and materials [59]. The list continues. Despite myr- iad applications and advancements, 3D printing faces challenges, e.g., topology, support, and surface optimization [60]. The remarkable thing is that porous structure, 3D printing, and fractal geometry have an intimate relationship. The description is as follows. Some porous structures used in biomedical, materials, and engineering applications cannot be fabricated without using additive manufacturing or 3D printing [61–65]. This fastens porous structure and 3D printing. Conducting experimental studies employing real porous structures (including cermet-based porous structures [41–44]) 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, “design for additive manufacturing” of porous structure is an important issue of research and implementation [60–62]. Developing methods, enabling design for additive Fractal Fract. 2021, 5, 40 3 of 19

Fractal Fract. 2021, 5, 40 3 of 19 Thus, “design for additive manufacturing” of porous structure is an important issue of research and implementation [60–62]. Developing methods, enabling design for addi- tivemanufacturing manufacturing of porous of porous structure, structure, fastens fastens fractal fractal geometry, geometry, porous porous structure, structure, and and 3D 3Dprinting. printing. The The explanation explanation is as is follows.as follows. In mostIn most cases, cases, porous porous structures structures to be to fabricated be fabri- catedby 3D by printing 3D printing are designed are designed using scaffold-based using scaffold-based approaches approaches [31,66–68 ].[31,66–68]. These structures These structuresexhibit regularly exhibit distributedregularly dist pores,ributed which pores, is notwhich the is case not inthe real case porous in real structures. porous struc- For tures.example, For considerexample, the consider cases shown the cases in Figures shown1 inand Figures2. In Figure 1 and1 ,2. a scaffold-basedIn Figure 1, a scaffold- porous basedstructure porous is shown structure [31]. is Particularly, shown [31]. Figure Particularly,1a shows Figure the unit 1a shows cell which the unit is used cell towhich create is usedthe scaffold to create (Figure the scaffold1b). Figure (Figure1c shows 1b). Figure the 3D 1c printed shows scaffoldthe 3D printed (porous scaffold structure), (porous and structure),Figure1d shows and Figure a magniﬁed 1d shows view a of magnified the printed vi porousew of the structure. printed Theseporous kinds structure. of structures These kindsdo not of mimic structures the real do porousnot mimi structures.c the real porous structures.

Figure 1. The 3D-printed scaffold-based porous structure: (a) unit-cell; (b) scaffold; (c) 3D-printed Figure 1. The 3D-printed scaffold-based porous structure: (a) unit-cell; (b) scaffold; (c) 3D-printed porous structure; and (d) the magnified view of the structure shown in (b). porous structure; and (d) the magniﬁed view of the structure shown in (b).

ForFor example, example, consider consider the the two two real real porous porous st structuresructures shown in Figure Figure 22.. The structure shownshown in in Figure Figure 22aa isis a clay pot used to grow indoorindoor plants.plants. The other structure shownshown inin FigureFigure 22bb is a ceramics-based coffee filter.filter. InIn bothboth casescases (Figure(Figure2 2),), the the pores pores have have random random sizessizes and areare distributeddistributed randomly, randomly, as as well. well. This This kind kind of structureof structure exhibits exhibits percolation percolation [69]. [69].As a As result, a result, fluids fluids and gasesand gases can pass can throughpass through the structure the structure as required. as required. Stochastic Stochastic point pointcloud-based cloud-based methods methods can create can create a porous a porous structure structure like the like real the ones real [ 61ones,62 ,70[61,62,70].]. Some Somesimple simple geometric geometric entities entities can fill can the fill spaces the spaces offered offered by the by randomly the randomly distributed distributed points pointsinside inside a boundary a boundary (e.g., (e.g., polyhedrons polyhedrons [62] [6 and2] and cylindrical cylindrical tentacles tentacles [70 [70]),]), leading leading to to a arealistic realistic porous porous structure. 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 Figure3. The structures shown in Figure3 were created using the system developed by Ullah et al. [ 62]. Figure3a presents the porous structure. Figure3b shows the representative cross-sections of the respective porous structures. Figure3c 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.

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(b)

(a)

(b)

Figure 2. Real porous structures: (a) a clay-based ceramics pot; and (b) a ceramics coffee filter. Figure (a) (right-hand-side figure) is reprinted with permission from 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 Figure 2. Real porous structures: (a) a clay-based ceramics pot; and (b) a ceramics coffee filter. Figure 2. Real porous structures:the den (asity) a clay-based and distribution ceramics of pot; points, and (b )the a ceramics underlying coffee porous ﬁlter. Figure structure (a) (right-hand-side exhibits a different Figure (a) (right-hand-side figure) is reprinted with permission from ref. [62]. Copyright 2021 ﬁgure) is reprinted with permissionporosity fromand distribution ref. [62]. Copyright of pores. 2021 ELSEVIER LICENSE. ELSEVIER LICENSE.

In stochastic point cloud-based porous structuring, the critical issue is how to generate(a) 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(b) 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.

(c)

(a)

FigureFigure 3. Point 3. Point cloud cloud-based-based porous porous structuring structuring:: (a) ( aporous) porous structures structures having having different different boundaries boundaries;; (b (b) )cross cross-sections-sections of of the respectthe respectiveive porous porous structures structures;; (c) cross (c)- cross-sectionssections when when the density the density of points of points increased increased by byeight eight times times.. Reprinted Reprinted with with per- missionpermission from ref. from [62 ref.].Copyright [62].Copyright 2021 2021 ELSEVIER ELSEVIER LICENSE LICENSE.. (b) 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

(c) 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. Section2 presents the settings to deﬁne an IFS fractal for generating stochastic point clouds. Section3 presents a mathemati- Figure 3. Point cloud-based porous structuring: (a) porous structures having different boundaries; (b) cross-sections of the respective porous structures; (c) cross-sections when the density of points increased by eight times. Reprinted with permission from ref. [62].Copyright 2021 ELSEVIER LICENSE.

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cal procedure to control the self-similarity levels in an IFS-fractal-generated point cloud. Section4 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 Sections3 and4.) The outcomes of Section3 are used in Section4. Section5 presents the solid CAD models and 3D-printed porous structures based on the outcomes of Section4. The contents presented in Section5 demonstrate the efﬁcacy of utilizing fractals in the modeling and 3D printing of realistic porous structures. Section6 concludes this study.

2. 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 [71–74]. Predefined probabilities control the frequency of participation of each map. To be more specific, we consider the following mathematical settings. Let {(xi, yi) | i = 1, ... , N} be the set of the recursively created point clouds where the initial point is (x0, y0) = (0, 0). The relationship between two consecutive points (xi−1, yi−1) and (xi, yi), i = 1, . . . , N, is as follows:

xi = ajxi−1 + bjyi−1 + ej, yi = cjxi−1 + djyi−1 + fj, ∃j ∈ {1, . . . , M} (1)

The vector of parameters (aj, bj, cj, dj, ej, fj) 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 [71–74]. In the mapping processes, multiple maps can be used. Let M be the number of maps to be used. While mapping (xi−1, yi−1) into (xi, yi), a map out of M maps can be selected randomly. The (random) selection process must be controlled. As such, probabilities pj ∈ [0, 1], ∀j ∈ {1, ... , M} are assigned to the maps so that p1 + ... + pM = 1. Thus, after performing the mapping process N times, the frequency of the j-th map participating in the mapping process must be about pj × N. To achieve this, ﬁrst, the cumulative probability of each map, denoted as cpj, is calculated as follows:

cpj = p1 + ... + pj (2)

The two consecutive cumulative probabilities produce an interval. This interval, in turn, determines the weight (wj) of the respective map. For the ﬁrst and last maps, the weights are w1 = [0, cp1) and wM = [cpM−1, cpM], respectively. For others, the weights are wj = [cpj−1, cpj), ∀j∈{2, ... , M − 1}. This means that the weights are mutually exclusive in- tervals that partition the interval [0, 1] so that {wj | j = 1, ... , M} = [0, 1] and wj ∩ wj + 1 = Ø, ∀j∈{1, . . . , M − 1}. Thus, the following relationship holds:

 [ ) =  0, cp1 , j 1 wj = [cpM−1, cpM = 1], j = M (3)  cpj−1, cpj otherwise

If a random number denoted as ri ∀ [0, 1] (i = 1, 2, ... ) belongs to wj, the corresponding map participates in the mapping process, others not. This ensures the relative frequencies of the respective maps according to pj 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. Fractal Fract. 2021, 5, 40 6 of 19

Algorithm 1: The main processes of the IFS Algorithm 1 The main processes of the IFS

1: N ∈ ℕ (iteration number)

2: M ∈ ℕ (number of maps) Define 3: {(aj, bj, cj, dj, ej, fj, pj)|j = 1, …, M} (affine maps)

4: (x0 = 0, y0 = 0) (initialize)

5: For j = 1, …, M

6: Calculate 𝑐𝑝 =𝑝 +⋯+𝑝

7: End For

8: w1 = [0, cp1), wM = [cpM−1,cpM = 1]

9: For j = 2, …, M – 1 Define 10: wj = [cpj−1, cpj)

11: End For

12: For i = 1, …, N

13: Generate a random number ri ∈ [0,1]

14: For j = 1, …, M

15: If ri ∈ wj Map 16: Then 𝑥 =𝑎𝑥 – +𝑏𝑦 +𝑒

17: 𝑦 =𝑐𝑥 – +𝑑𝑦 +𝑓

18: End For

19: End For

20: Define PC = {(xi, yi)|i = 0, …, N} (output)

The point cloud denoted as PC produced by the IFS Algorithm is the stochastic point cloud representing the given fractal shape. For example, Figure4 shows nine point-clouds representing nine different fractal shapes (tree, seahorse, fern leaf, snowﬂakes, and the Sierpinski Carpet). The settings of the afﬁne maps {(aj, bj, cj, dj, ej, fj, pj) | j = 1, ... , M} can be found in [71–74]. The Sierpinski Carpet (Figure4i) is considered in this study to model a porous structure. As far as manufacturing is concerned, the IFS Algorithm generated Fractal Fract. 2021, 5, 40 7 of 19

Fractal Fract. 2021, 5, 40 7 of 19 Fractal Fract. 2021, 5, 40 point cloud must undergo some transformation, controlling the self-similarity levels.7 of 19 For example, consider the case in Figure5. Figure5a shows a point cloud of fern leaf (same as Figure4c). Figure5b shows the same shape; this time, the self-similarity is controlled up to levelto level three. three. The The self-similarity self-similarity level level controllingcontrolling mechanism mechanism employs employs a set a set of ofone one-to-one-to-one to level three. The self-similarity level controlling mechanism employs a set of one-to-one maps.maps. Sometimes, selected selected affine afﬁne maps maps out outof M of mapsM maps can be can omitted be omitted for the forsake the of con- sake of maps. Sometimes, selected affine maps out of M maps can be omitted for the sake of con- controllingtrolling the thelevel level of self of- self-similarity.similarity. See [72,73] See [72 for,73 the] for details. the details. It is worth It is mentioning worth mentioning that trolling the level of self-similarity. See [72,73] for the details. It is worth mentioning that thatself- self-similarsimilar structures structures created created by tree by-like tree-like fractals fractalshave been have used been to reinforce used to a reinforce 3D printed a 3D self-similar structures created by tree-like fractals have been used to reinforce a 3D printed printedstructure structure [75,76]. [In75 ,this76]. case, In this solid case, models, solid models,not the notpoint the cloud, point of cloud, the self of- thesimilar self-similar seg- 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 segmentsments are areused, used, which which is not is the not focus the focus in this in study. this study. However, However, the manufacturability the manufacturability of these self-similar segments limits the level up to which the self-similar tentacles can be ofthese these self self-similar-similar segments segments limits limits the thelevel level up upto which to which the theself self-similar-similar tentacles tentacles can canbe be fabricated with required accuracy [72,73,77]. fabricatedfabricated with with required required accuracy accuracy [72 [72,73,77,73,77]. ].

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Figure 4. Nine point-clouds showing some well-known fractals: (a) tree; (b) seahorse; (c) fern leaf; FigureFigure 4. Nine point point-clouds-clouds showing showing some some well well-known-known fractals fractals:: (a) tree (a);tree; (b) seahorse (b) seahorse;; (c) fern (c )leaf fern; leaf; (d–h) snowflakes; (i) the Sierpinski Carpet. Reprinted with permission from ref. [72]. Copyright ((dd––hh)) snowﬂakes;snowflakes; ( i)) the the Sierpinski Sierpinski Carpet Carpet.. Reprinted Reprinted with with permission permission from from ref. ref. [72 [72]. Copyright]. Copyright 2021 2021 Fuji Technology Press Ltd. (Tokyo, Japan). Fuji2021 Technology Fuji Technology Press P Ltd.ress Ltd (Tokyo,. (Tokyo Japan)., Japan).

(a) (b) (a) (b)

Self-similarity level Selfc-ontrolsimilarity process level control process

Figure 5. Concept of controlling self-similarity levels of a fractal: (a) original point cloud; and (b) Figure 5. Concept of controlling self-similarity levels of a fractal: (a) original point cloud; and (b) Figurepoint cloud 5. Concept controlled of controlling up to self-similarity self-similarity level levelsthree. of a fractal: (a) original point cloud; and ( point cloud controlled up to self-similarity level three. mboxtextbfb) point cloud controlled up to self-similarity level three.

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3. 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 = {(xsi, ysi)|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 PCG−1 = {(xG−1,i, yG−1,i) | i = 0, 1, . . . , N}. The following one-to-one map is applied to each point in PCS to get its counterpart in PCG−1. Thus, the following relationship holds:

xG−1,i = acxsi + bcysi + ec, yG−1,i = ccxsi + dcysi + fc, ∀i ∈ {0, . . . , N} (4)

The mapping parameters (ac, bc, cc, dc, ec, fc) 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 PCG−2 = {(xG−2,i, yG−2,i) | i = 0, 1, . . . , 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 PCG−1 to get its counterpart in PCG−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 (xG−1,i, yG−1,i) to (xG−2,i, yG−2,i). 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 PCG−3, PCG−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 = {(xF,k, yF,k) | k = 0, ... , L} and fills the empty regions PCG−1, PCG−2, PCG−3, ... , following the mapping processes defined in Step 2,3, ... , respectively. Let PCF0 be the point clouds generated in this step. Algorithm 2: IFS-inspired mapping Algorithm 2 IFS-inspired mapping

1: For i = 1, …, N

2: Generate a random number ri ∈ [0, 1] 3: For j = 1, …, M

4: If ri ∈ wj Map 5: Then 𝑥, =𝑎𝑥, +𝑏𝑦, +𝑒

6: 𝑦, =𝑐𝑥, +𝑑𝑦, +𝑓 7: End For 8: End For

0 A point cloud consisting of PCG−1, PCG−2, ... , and PCF can be used to model a porous structure. Nevertheless, the above steps are the general procedure for creating a model of a porous structure. The sequences of the steps can be designed based on the need Fractal Fract. 2021, 5, 40 9 of 19

A point cloud consisting of PCG−1, PCG−2, …, and PCF′ can be used to model a porous structure. Nevertheless, the above steps are the general procedure for creating a model of a porous structure. The sequences of the steps can be designed based on the need 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.

Fractal Fract. 2021, 5, 40 4. Modeling Porous Structures Using Sierpinski Carpet 9 of 19 This section shows how to modify the Sierpinski Carpet-driven point cloud (an IFS fractal,of the modeler. shown For in example, Figure a modeler 4i) so canthat apply it becomes Step 1, Step a 2, Stepporous 5, Step structure 3, Step 4, Step model. The steps pre- sented4, and Step in 4the successively. previous Another section modeler will canbe applyused Step to achieve 1, Step 2, Stepthis 3, goal. Step 4, Step 5, successively.First, consider This means the that settings depending of on the the needIFS fractal or preferences called of the a model, Sierpinski the steps Carpet. The settings and their sequence can be determined. There is no restriction as such. of the Sierpinski Carpet [74] are listed in Table 1. As listed in Table 1, the Sierpinski Carpet needs4. Modeling eight Porous affine Structures maps with Using equal Sierpinski weights Carpet (probability) of 1/8. The maps create a point cloudThis that section resides shows in how a square to modify-shaped the Sierpinski boundary Carpet-driven given point by cloudvertices (an IFS(0, 0) (1, 0), (1, 1), and fractal, shown in Figure4i) so that it becomes a porous structure model. The steps presented (0,in the1). previousThe distri sectionbution will be of used points to achieve creates this self goal.-similar squares, as shown in Figure 6. As seen in FigureFirst, consider 6, the the self settings-similarity of the IFS is fractal scaled called by the 1/3. Sierpinski The Carpet. maps The symmetrically settings distribute the pointsof the Sierpinski around Carpet the [scaled74] are listed squares. in Table As1. As seen listed in in TableFigure1, the 6, Sierpinski the self Carpet-similar levels above the needs eight afﬁne maps with equal weights (probability) of 1/8. The maps create a point tcloudhird thatlevel resides are not in a square-shapedvisible. If more boundary points given are by injected vertices (0, (i.e., 0) (1, N 0), is (1, increased), 1), and (0, the higher levels might1). The distributionbe visible. of This points issue creates is self-similar, however, squares, beyond as shown the inscope Figure of6. Asthis seen study. in Figure6, the self-similarity is scaled by 1/3. The maps symmetrically distribute the points around the scaled squares. As seen in Figure6, the self-similar levels above the third level Table 1. Settings of the Sierpinski Carpet [74]. 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. Affine Maps Table 1. SettingsParameters of the Sierpinski Carpet [74]. j = 1 Afﬁne2 Maps 3 4 5 6 7 8 Parameters aj 1/3 1/3j = 1/3 1/3 1/3 1/3 1/3 1/3 1 2 3 4 5 6 7 8 bj 0 0 0 0 0 0 0 0 aj 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 cj 0 0 0 0 0 0 0 0 bj 0 0 0 0 0 0 0 0 cj d0i 0 01/3 01/3 01/3 01/3 01/3 0 1/3 1/3 1/3 di 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 ej 0 0 0 1/3 1/3 2/3 2/3 2/3 ej 0 0 0 1/3 1/3 2/3 2/3 2/3 fj f0j 1/3 2/30 01/3 2/32/3 00 1/32/32/3 0 1/3 2/3 pj 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8 pj 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8

Third level Second level

First level

Zero level

FigureFigure 6. 6.Sierpinski Sierpinski Carpet Carpet and its self-similarityand its self- nature.similarity nature.

Let us apply Steps 1, . . . ,5 and modify the point cloud in Figure6. Let us apply Steps 1, …,5 and modify the point cloud in Figure 6.

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Step 1 is applied as follows: the seed point cloud (PCS) is created using the zero- level shape (Figure6). The zero-level shape consists of lines P1P2, P2P3, P3P4, and P4P1, where P1 = (0, 0), P2 = (1, 0), P3 = (1, 1), and P4 = (0, 1), as shown in Figure7a. As such, PCS = {(xsi, ysi)|i = 0, 1, . . . , N} consists of the points on the lines P1P2, P2P3, P3P4, and P4P1. 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 PCG−1 = {(xG−1,i, yG−1,i) | i = 0, 1, ... , N}. For this, the following formulation is used:

1 1 1 1 x − = x + , y − = y + , ∀i ∈ {0, . . . , N} (5) G 1,i 3 si 3 G 1,i 3 si 3

As such, the values of the critical parameters are ac = 1/3, bc = 0, cc = 0, dc = 1/3, ec = 1/3, and fc = 1/3. The point clouds PCS and PCG−1 are shown in Figure7b. Step 3 is applied as follows. The affine maps shown in Table1 are used to create the second-generation modified point cloud PCG−2 from PCG−1, as defined in Algorithm 2. The point clouds PCS, PCG−1, and PCG−2 are shown in Figure7c. Step 4 is applied as follows. This time, it is applied thrice. The first execution of Step 4 on the point cloud PCG−2 results in the third level self-similar squares given by the point cloud PCG−3. The point clouds PCS, PCG−1, PCG−2, and PCG−3 are shown in Figure7d . The second execution of Step 4 on the point cloud PCG−3 results in the fourth level self- similar squares given by the point cloud PCG−4. The point clouds PCS, PCG−1, PCG−2, PCG−3, and PCG−4 are shown in Figure7e. The third execution of Step 4 on the point cloud PCG−4 results in the fifth level self-similar squares given by the point cloud PCG−5. The point clouds PCS, PCG−1, PCG−2, PCG−3, PCG−3, PCG−4, and PCG−5 are shown in Figure7f. 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, Figure8 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 Figure7. As seen in Figure8, 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 = {(xF,k, yF,k) | k = 0, 1, ... , L} is a random point cloud in the interval of [0, 1]. Thus, xF,k ∈ [0, 1] and yF,k ∈ [0, 1], ∀k ∈ {0, . . . , L}. One of the examples of PCF is shown in Figure9a. This point cloud is mapped using the procedure underlying Steps 2–4, where the affine maps of the Sierpinski Carpet are used. The resulting point cloud (PCF0) is shown in Figure9b. As seen in Figure9b , this time, the level-controlled pores are filled by points. Therefore, this point cloud (PCF’) can be added to the level-controlled point cloud if the pores are not desired. Fractal Fract. 2021Fractal, 5, 40 Fract. 2021, 5, 40 11 of 19 11 of 19

(a) (b) 1 1 P2 P3 P2 P3

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

P1 P4 P1 P4 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x (c) (d) 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x (e) (f) 1 1

0.8 0.8

0.6 0.6

y 0.4 0.4

0.2 0.2

0 0 0.2 0.4 0.6 0.8 1 0 x 0 0.2 0.4 0.6 0.8 1 x Figure 7. Level-controlledFigure 7. Level point-controlled clouds ofpoint the clouds Sierpinski of the Carpet: Sierpinski (a) seed Carpet point: (a cloud;) seed point (b) point cloud cloud; (b) ofpoint (a) and the ﬁrst level point cloud;cloud (c )of point (a) and clouds the first of (b level) and point the second cloud; level(c) point point clouds cloud; of ( d(b)) point and the clouds second of ( clevel) and point the third cloud level; point (d) point clouds of (c) and the third level point cloud; (e) point clouds of (d) and the fourth level cloud; (e) point clouds of (d) and the fourth level point cloud; and (f) point clouds of (e) and the ﬁfth level point cloud. point cloud; and (f) point clouds of (e) and the fifth level point cloud.

Fractal Fract.Fractal 2021 Fract. ,2021 5, 40, 5, 40 12 of 19 12 of 19

1.2

Fractal Fract. 2021, 5, 40 12 of 19 1

4-th level 1.2 3-rd level 0.8 2-nd level 1-st level 1 zero-level 0.6

4-th level

y 3-rd level 0.8 2-nd level 1-st level 0.4 zero-level 0.6

y 0.4 0.2

0.2 0

0 -0.2 -0.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 0.6 0.8 x 1 1.2 x Figure 8. Level-controlled point clouds of the Sierpinski Carpet up to the fourth level generated from Figure 8. Level8. Level-controlled-controlled point clouds point of theclouds Sierpinski of the Carpet Sierpinski up to the fourthCarpet level up generated to the fourth level generated a relatively small seed point cloud. from a relativelya relatively small small seed point seed cloud. point cloud.

(a) (b) (a) (b)

FigureFigure 9.9. FillingFilling porespores usingusing aa randomrandom fillerfiller point point cloud: cloud: ( a(a)) a a random random filler filler point point cloud; cloud and; and ( b(b)) the the fifth level point cloud after filling operation by filler point cloud (a). fifth level point cloud after filling operation by filler point cloud (a).

5.5. SolidSolid ModelingModeling andand 3D3D PrintingPrinting ThisThis section shows shows some some solid solid models models of porous of porous structures, structures, as well as as well the as3D theprinted 3D printedstructures. structures. FigureFigureFigure 9. Filling 10 showsshows pores the the point pointusing clouds clouds a random controlled controlled filler up uptopoint the to fourth thecloud fourth level: (a) leveland a random its and solid its model.filler solid point cloud; and (b) model.Thethe pointfifth The levelclouds point point cloudsused incloud used this case inafter this are casefilling shown are operation shownin Figure in Figure10 bya. Thefiller 10 solida. point The CAD solid cloud model, CAD (a model,shown). shownin Figure in Figure10b, is 10createdb, is created using commercially using commercially available available software software that can that convert can convert a raster a rastergraphics5. Solid graphics entity Modeling entity into a into vector and a vector graphics 3D graphics Printing entity entity using using voxel voxel-based-based geometric geometric modeling. modeling. Fig- Figureure 11 11showsshows another another similar similar case. case. This This time, time, the thepoint point clouds, clouds, shown shown in Figure in Figure 11a, 11con-a, controlledtrolledThis up upto section tothe the fifth ﬁfth levelshows level is isconsidered. considered.some solid The The modelscorresponding corresponding of porous solid solid CAD structures, model isis shownshown as well as the 3D printed instructures. 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. 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. Fig- ure 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.

Fractal Fract. 2021, 5, 40 13 of 19

FractalFractal Fract. Fract. 2021 ,2021 5, 40, 5, 40 13 of 1319 of 19

in Figure 11b. This way, a user can apply the procedure described in Sections3 and4 to digitize a porous structure model given the level-controlled point clouds of the IFS fractal.

(a) (a) (b) (b)

FigureFigure 10.Figure Point10. Point 10. clouds Point clouds controlled clouds controlled up controlled to the up fourth to up the level to fourth the and fourth its level solid level CADand itsand model: solid its (solida CAD) the pointCAD model clouds; model: (a) andthe: (a) (pointb the) the point solid CADclouds model.clouds; and; and(b) the (b) solid the solid CAD CAD model. model.

(a) (a) (b) (b)

FigureFigure 11.Figure Point11. Point 11. clouds Point clouds controlled clouds controlled up controlled to the up ﬁfth to up level the to andfifththe its fifthlevel solid level and CAD itsand model: solid its solid (aCAD) the CAD pointmodel model clouds;: (a) the: ( b(a)) thepoint the solid point CAD model.cloudsclouds; (b) ;the (b) solid the solid CAD CAD model. model.

To obtain more insightsTo obtain into more the insights fabricatio into then of fabrication porous of structures, porous structures, a commercially a commercially To obtain availablemore insights 3D printer into is used. the Several fabricatio 3D-printedn of porous structures structures, have been produced. a commercially Three of availableavailable 3D printer3D printerthe is resultsused. is used. areSeveral shown Several 3D in- this printed3D section.-printed structures Figure structures 12 showshave have been the pointbeen produced. cloudsproduced. controlled Three Three of up of to the resultsthe results are shownare levelshown in ﬁve, this in the thissection. corresponding section. Figure Figure 3D-printed 12 shows 12 shows structure, the pointthe and point aclouds magniﬁed clouds controlled view controlled of the up structure. toup to levellevel five, five, the correspondingthe correspondingAs seen in Figure 3D- printe3D 12,- theprinted porous structure,d structure, structure and exhibits anda magnified a amagnified complex view network view of the amongof structure.the thestructure. pores in the structure. This is partly because of the limitation of the solid CAD model and partly As seenAs seen in Figure in Figure 12, the 12, porousthe porous structure structure exhibits exhibits a complex a complex network network among among the poresthe pores in in because of the limitation of the 3D printer. the structure.the structure. This This is partly is partly because because of the of limitationthe limitation of the of solidthe solid CAD CAD model model and andpartly partly becausebecause of the of limitationthe limitation of the of 3Dthe printer.3D printer.

(a) (b) (a) (b) (c) (c)

Fractal Fract. 2021, 5, 40 13 of 19

(a) (b)

Figure 10. Point clouds controlled up to the fourth level and its solid CAD model: (a) the point clouds; and (b) the solid CAD model.

(a) (b)

Figure 11. Point clouds controlled up to the fifth level and its solid CAD model: (a) the point clouds; (b) the solid CAD model.

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 Fractal Fract. 2021, 5, 40 the structure. This is partly because of the limitation of the solid CAD model14 of 19and partly because of the limitation of the 3D printer.

(a) (b) (c)

Fractal Fract. 2021, 5, 40 14 of 19

Figure 12. Point clouds controlled up to the fifth level and the 3D-printed porous structure: (a) point clouds; (b) 3D-printed porous structure; and (c) a magnified view of the printed structure. Figure 12. Point clouds controlled up to the fifth level and the 3D-printed porous structure: (a) point clouds; (b) 3D-printed porous structure; and (c) a magnifiedFigure 13 view shows of the printedthe point structure. clouds controlled up to level five, the corresponding 3D- printed structure, and a magnified view of the structure. This time, the filling operation Figure 13 shows the point clouds controlled up to level five, the corresponding 3D- was carriedprinted structure,out, as descried and a magnified in the previous view of the section. structure. Thus, This in time, the the point filling clouds operation (Figure 13a), the firstwas and carried second out, as level descried self in-similar the previous square section. regions Thus, are in the not point filled clouds with(Figure points. 13a), The correspondingthe first porous and second structure level self-similar (Figure 13 squareb,c) regionsexhibits are a notcomplex filled with network points. Theamong corre- pores. As seen insponding Figure porous13, the structure porous (Figurestructure 13b,c) exhibits exhibits a a complex complex networknetwork among among pores. the Aspores in the structure.seen in Like Figure the 13 previous, the porous case, structure this exhibits is partly a complex because network of the among limitation the pores of inthe the solid CAD structure. Like the previous case, this is partly because of the limitation of the solid CAD modelmodel and andpartly partly because because of of the the limitationlimitation of of the the 3D printer.3D printer.

(a) (b) (c)

Figure 13.Figure Point 13.cloudsPoint cloudscontrolled controlled up upto tothe the fifth fifth level with with a filling a filling operation operation and the and 3D-printed the 3D porous-printed structure: porous (a) point structure: (a) point cloudsclouds;; (b) (3Db) 3D-printed-printed porous porous structure; structure and; and (c) a ( magnifiedc) a magnified view of view the printed of the structure. printed structure.

What if the structure is scaled down while printing it? When the structure is produced Whatby reducing if the the structure scale, the poreis scaled network down becomes while more printing complex. it? As When a result, the randomness structure is pro- ducedin by the reducing pore sizes increases.the scale, This the is becausepore network of the limitation becomes of the more 3D printer. complex. For example,As a result, ran- domnessconsider in the the pore case shownsizes increases. in Figure 14 This. The is point because clouds of shown the limitation in Figure 13 of are the used 3D toprinter. For example,print consider the structure the shown case inshown Figure in 14 Figure. The structure 14. The is scaledpoint downclouds by shown half during in Figure the 13 are used printing,to print resultingthe structure in a more shown complex in Figure network 14. among The thestructure pores than is scaled that of Figuredown 13 by. half during the printing, resulting in a more complex network among the pores than that of Figure 13.

(a) (b)

Figure 14. 3D-printed porous structure: (a) 3D-printed porous structure; and (b) a magnified view of the printed structure.

Fractal Fract. 2021, 5, 40 14 of 19

Figure 12. Point clouds controlled up to the fifth level and the 3D-printed porous structure: (a) point clouds; (b) 3D-printed porous structure; and (c) a magnified view of the printed structure.

Figure 13 shows the point clouds controlled up to level five, the corresponding 3D- printed 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.

(a) (b) (c)

Figure 13. Point clouds controlled up to the fifth level with a filling operation and the 3D-printed porous structure: (a) point clouds; (b) 3D-printed porous structure; and (c) a magnified view of the printed structure.

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 Fractal Fract. 2021, 5, 40 example, consider the case shown in Figure 14. The point clouds shown in Figure 13 are 15 of 19 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.

(a) (b)

Fractal Fract. 2021, 5, 40 15 of 19

Figure 14. 3D-printed porous structure: (a) 3D-printed porous structure; and (b) a magnified view of the printed struc- Figure 14. 3D-printed porous structure: (a) 3D-printed porous structure; and (b) a magniﬁed view of the printed structure. ture. In addition to using a voxel-based solid modeling of porous structure from level- controlledIn addition point to clouds using of a voxel-basedIFS fractals, other solid solid modeling modeling of approaches porous structure can be used. from In level- controlledthis respect, point one clouds of the of obvious IFS fractals, modeling other approaches solid modeling is to model approaches each point can of be the used. pointIn this

respect,clouds one by of using the obvious special modeling polyhedrons approaches (e.g., prismoid). is to model For example, each point consider of the the point case clouds by usingshown special in Figure polyhedrons 15. In this case, (e.g., the prismoid). modeling concept For example, is to model consider each point the of case an shownIFS- in generated point cloud (in this case, a point cloud representing Sierpinski Carpet created Figure 15. In this case, the modeling concept is to model each point of an IFS-generated by the IFS Algorithm defined in Algorithm 1) by a hexagonal cylinder (see Figure 15a). pointThis cloud way, (in multipl this case,e point a clouds point cloudcan be representingcreated and solid Sierpinski modeled. CarpetThe point created clouds bylook the IFS Algorithmalike but deﬁned are not the in same Algorithm due to the 1) bystochasticity a hexagonal involved cylinder in point (see cloud Figure creation 15a). process This way, multiple(see IFS point Algorithm clouds 1 can). In beFigure created 15b, andthree solid point modeled.clouds are Theshown point to be clouds modeled look by alikethe but are notproposed the same concept. due The to the solid stochasticity models of these involved point clouds in point are shown cloud in creation Figure 15 processc. Finally, (see IFS Algorithmthese solid 1). Inmodels Figure can 15 beb, combined three point to print clouds a porous are shown structure. to One be modeled printed porous by the struc- proposed concept.ture is The shown solid in Figure models 15d of. these point clouds are shown in Figure 15c. Finally, these solid modelsIn synopsis, can be IFS combined-generated to stochastic print a porous point clouds structure. provide One a flexible printed means porous to model structure is shownporous in Figure structures, 15d. and thereby, to fabricate them using 3D printing or additive manufacturing.

60 60 60 55 55 55 50    50     50       45  45  45                   40    40        40           35   35  35                30    30    30   (a)          25     25      25                    (b) 20  20  20              15   15 15              10    10    10                5    5    5               0 0   0     -5 -5 -5 -10 -10 -10 -10 0 10 20 30 40 50-10 60 0 10 20 30 40 50-10 60 0 10 20 30 40 50 60

(c)

(d)

Figure 15. Prismoid-based solid modeling of IFS fractal generated point clouds and a 3D printed porous structure: (a) Figure 15. Prismoid-based solid modeling of IFS fractal generated point clouds and a 3D printed porous structure: modeling concept; (b) three different point clouds generated by the IFS Algorithm; (c) prismoid-based solid models of the (a) modelingrespectiveconcept; point ( bclouds) three; and different (d) 3D-printed point clouds porous generatedstructure of bycombining the IFS solid Algorithm; models (inc )(c prismoid-based). solid models of the respective point clouds; and (d) 3D-printed porous structure of combining solid models in (c). 6. 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.

Fractal Fract. 2021, 5, 40 16 of 19

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.

6. 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 ob- served in the natural world, living organisms, and artificial objects (particularly in micro- nano 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 cermet- based 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.

Author Contributions: Conceptualization, A.S.U., D.M.D., Y.S., S.Y. and A.K.; methodology, A.S.U., D.M.D. and Y.S.; mathematical formulation, A.S.U.; modeling A.S.U., Y.S. and S.Y.; resources, A.S.U. and A.K.; 3D printing and data curation, A.S.U., Y.S., S.Y. and A.K.; writing—original draft prepa- ration, A.S.U. and D.M.D.; writing—review and editing, A.S.U., D.M.D., Y.S. and S.Y.; supervision, A.S.U. and D.M.D. All authors have read and agreed to the published version of the manuscript. Fractal Fract. 2021, 5, 40 17 of 19

Funding: This research received no external funding. Data Availability Statement: The data are available from the corresponding author upon request. Acknowledgments: Authors gratefully acknowledge Shota Yamamoto and Ryo Yamauchi for their assistances in modeling building and 3D printing activities. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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