3D Printing of Components with Tailored Properties Through Hilbert Curve Filling of a Discretized Domain

Abstract

Owing to the localized line-by-line and layer-by-layer style of material deposition, 3D printing remains an ideal candidate for fabrication of components with tailored properties (also referred to as functionally gradient components). The present work tries to exploit this advantage, in the extrusion-based 3D printing process, to fabricate components with varying set of properties at different locations. The implementation is done using Hilbert area-filling curves with the displacement per unit force (i.e., compliance) applied being the property varying in a gradient manner. Four input parameters have been considered to study their effect on the compliance, and the single most influencing parameter has been selected using analysis of variance (ANOVA) for further study. Mapping of a selected input variable on the desired property has been discussed through numerical and experimental tests. Based on these studies, a demonstrative case study of a shoe sole has been designed and fabricated. Deflections of the fabricated component have been measured at different locations for uniform loading conditions. The deflection behavior of the fabricated component is found to be in line with the gradient force response required, thus validating the proposed approach. The current study is intended to provide the basic framework for fabrication of components tailored for force response using Hilbert curves.

Introduction

Although anisotropy is commonly mentioned as a limitation of 3D printing, the same becomes an advantage in fabrication of heterogeneous components with tailored properties. Functionally graded materials (FGMs), as these types of structures are generally called, have received significant attention in many engineering and biomedical fields.^1,2 Lightweight components with variable properties are also a common requirement for biomedical applications such as prosthetics.

The ideal processes for fabrication of FGMs are those that allow multimaterial addition such as direct energy deposition (DED) due to the ease of multimaterial feed control. FGM alloys were fabricated by various researchers with different compatible materials such as Ti-V, Ti-Ta, Ti-Mo, and Ti-Nb using the laser-based DED technique, with the microstructure and mechanical property being the gradient entities that had been accomplished without defects.^1,3,4

In processes such as powder bed fusion and selective laser sintering, where a layered raw material is acted upon by a moving energy source such as laser, it is possible to make FGMs by varying the process parameters such as laser power at each location. Geng and Harrison demonstrated the feasibility of such FGMs through modeling of Ti6Al4V.⁵ It is also possible to get gradient properties either by the use of multiple materials across each layer (as demonstrated by Chung and Das for selective laser sintering⁶) or by multiple materials in a given layer itself (as demonstrated by Huang et al. for stereolithography⁷).

Although the material extrusion method is very popular and extensively used to fabricate polymer components, its use to fabricate FGM components has been limited. Ren et al. fabricated the gradient matrix through a digital feed of multiple materials into a single converging extruder.⁸ Zheng et al. used a similar multifeed extrusion setup combined with UV curing for silicone elastomers.⁹ Wang et al. fabricated functionally gradient, thermoplastic composite parts with anisotropic, thermal conductive properties by modifying the feeding system to mix multiple materials before extrusion.¹⁰ The use of materials such as carbon fibers¹¹ or variable elasticity polymers with directional properties¹² is another possible method.

While the above-discussed processes present various means of fabricating FGMs, they require a specific setup and materials. It would be ideal if the tailored properties can be realized by varying the internal infill structure of the same material as this technique would apply to any existing setup without significant changes. This can be achieved by varying the infill parameters such as infill pattern,^13,14 infill density,¹³ and raster angle.¹⁵ Fractal infill patterns have been of particular value in this context as they facilitate the synergizing of global and local properties of the infill. A direct fabrication method for such fractal objects using the iterative function system was proposed by Chiu et al.; this approach avoids relying on the Euclidean geometry representation for tool path generation.¹⁶ Larimore et al. used space-filling curves based on meandering lines with equal lengths in the x and y directions to fabricate graded dielectric structures.^17,18 Liu et al. used the moving morphable component and void methods to develop the design of shell-graded–infill structures; this is particularly useful in the design of coated structures.¹⁹ Hilbert is another infill form explored by researchers due to its continuity, facilitating single continuous motion of the head.²⁰ Kumar et al. studied five types of space-filling fractal curves, viz, e-curve, Hilbert, Peano, Macrotile 3 × 3, and Macrotile 4 × 4 curves, for objects with controlled porosity.²¹ These find specific application in fabrication of personalized bone tissue engineering scaffolds with controlled architecture.²² Kapil et al. used a hybrid of zigzag and fractal curves to obtain continuous deposition paths.²³ These studies demonstrate the potential of space-filling curves in controlling the properties of fabricated components. The current study presents the use of fractal infill patterns in a fused filament fabrication setup for obtaining gradient properties and builds upon the previous work of the authors for fabricating gradient materials joined by an elastomer film.²⁴

The objective of the current work is to evolve a methodology for fabricating components with a digitally structured deflection at various locations for a given force. This is achieved with the help of Hilbert-based fractal area filling. The current work is organized as follows: the Realizing Tailored Properties section describes various types of area filling techniques to achieve tailored properties. The selection of the most influential parameter to achieve the desired magnitude of deflections has been discussed in the Identification of Influential Parameters section. Furthermore, the numerical and experimental analysis for mapping the control variable with the desired parameter is presented in the Mapping of the Control Variable with the Desired Parameter section. The Illustrative Case Study section presents the fabrication and validation of an illustrative case study of gradient foot pressure. Finally, the Conclusions section summarizes the current work based on observed results.

Realizing Tailored Properties

Identification of a suitable infill pattern

Various types of infill patterns are used for 3D printing, with concentric, rectilinear, honeycomb, and Hilbert curve patterns being the most common. Nonperiodic infill patterns, such as rectilinear and concentric patterns, are geometry specific and global in nature due to which a consistent property can be obtained throughout the structure. Periodic infill structures such as honeycomb and Hilbert curve, on the other hand, are independent of geometry and it is possible to control the size locally due to which the gradient property can be achieved. Figure 1 illustrates these infill patterns along with their variable density counterparts. Rectangular and honeycomb infill patterns have a discontinuity at the interface of the gradient infill density, while the concentric infill pattern has a stability problem because of the disconnected path with the adjacent loop, as can be inferred from Figure 1. Hence, these infill patterns are not suitable for a continuously changing gradient, making the Hilbert curve the best curve for such applications. Localized control of the infill density and continuous nature of the Hilbert curve even for multiple infill densities make that possible. Owing to these advantages, the Hilbert curve was adopted for implementation of the gradient property.

FIG. 1.

Illustration of different infill patterns for area-filling. Color images are available online.

Generation of Hilbert infill pattern

The Hilbert curve is a continuous area-filling curve generated by traversing every point of a $2^n\times 2^n$ grid with a self-replicating pattern.^25,26 Here, n represents the order of the curve. The illustrative image of the Hilbert curve of different orders has been shown in Figure 2. The n + 1th order of the Hilbert curve is formed by arranging the nth-order pattern in the four sections. The top left and right places of the n + 1th-order pattern have been arranged without any rotation (i.e., as it is or with zero rotation), while the bottom left and right places are arranged by rotating the nth-order Hilbert pattern 90° in clockwise and anticlockwise directions, respectively, as shown in Figure 2. Now, these four disjoint curves have been connected with a straight line to form the n + 1th order of the Hilbert curve. These area-filling curves have a continuous tool path that means one start and one endpoint without intersecting any lines. Hence, these structures have the additional capability for more elastic deformation due to their continuous infill path.

FIG. 2.

Formation of the Hilbert curve. Color images are available online.

Role of loading direction in discretization

For a given component, the direction in which the load is applied (and the direction in which gradient deflection behavior is expected) can be simplified into two categories: parallel and perpendicular to the build direction. The unit slice resulting from each of these possibilities is shown in Figure 3. This unit slice and the implied infill path are then discretized for generating the tailored response as needed. The case of parallel loading (i.e., loading direction parallel to the build direction) has been evaluated in earlier published works and hence not duplicated in this work,²⁴ with focus restricted to perpendicular loading.

FIG. 3.

Unit cell formed based on the loading direction. Color images are available online.

In the case of parallel loading, the discretization domain coincides with the infill direction, making it possible to have a continuous Hilbert space. However, in reality, such a slice alone is inadequate for loading due to material discontinuity. Hence, parallel direction discretization is not very helpful when force–deflection gradient behavior is needed; it is nevertheless apt for obtaining gradient physical properties such as thermal conductivity.

Perpendicular loading (i.e., loading direction perpendicular to the build direction) ensures a tailored response when a load is applied, making it better suited for fabrication of force–deflection gradient structures. In perpendicular loading, the resulting unit cell, as seen in the Figure 3, will be a bounded cell. Conceptually, this bounded unit cell can also accommodate other forms of infill patterns. However, to retain the possibility of combining different loading directions, Hilbert infill is chosen. The generation and fabrication of parts using various input parameters related to the Hilbert curve cells in perpendicular direction have been discussed subsequently.

Identification of Influential Parameters

Experimental plan

The deflection at a given location is dependent on various process parameters selected. Among them, only parameters related to the infill pattern are of interest to us here; machine-specific parameters such as nozzle diameter and layer height, although having an influence, are maintained constant in this study. Hence, the following four parameters have been taken as input variables for the process. For each of these variables, three settings as described below have been selected:

• 1.Order of the Hilbert curve: The order of the Hilbert curve refers to the number of grid points generated through the $2^n\times 2^n$ grid, with n being the order of the curve. Three different orders of the Hilbert curve have been considered, 2nd, 3rd, and 4th orders.
• 2.Grid spacing: Grid spacing is the gap between two nodes of a fully covered area and provides for finer density control of the design space. As the 4th order is the highest order selected in these experiments, the grid spacing has been defined with reference to it and selected in multiples of extrusion width. Hence, the values selected for grid spacing are 0.8, 0.9, and 1.0 mm, representing 2w, 2.25w, and 2.5w, where w is the extrusion width (w = 0.4 mm). Accordingly, the unit cell size of the samples will be 16 times the grid spacing (12.8, 14.4, and 16.0 mm, respectively). It can be noted that this relationship between unit cell size and grid spacing is also a crucial consideration during discretization of the design domain.
• 3.Horizontal span: This refers to the number of unit cells in the direction perpendicular to the loading direction. For ease of reference, the loading direction is maintained vertically downward, making this parameter the horizontal span; 1, 2, and 3 are the number of unit cells thus considered.
• 4.Vertical span: The number of unit cells in the direction parallel to the loading direction, with 1, 2 and 3 being the values selected.

In the above parameters, order and grid space indicate the unit cell shape and density, while horizontal and vertical spans indicate the stretch of these cells. These parameters and their selected values are tabulated in Table 1. Extrusion width, layer height, and number of layers have been maintained constant. Extrusion width (w) and layer height (h) were selected to be 0.4 and 0.3 mm, respectively. All samples were fabricated for 20 layers, with the height reaching 6.0 mm.

Table 1.

Input Parameters Considered During Fabrication of the Samples

1	Order of the Hilbert curve (O)	:
		O2: 2nd order	O3: 3rd order	O4: 4th order
2	Grid spacing (S)	:
		S0.8 (2w)	S0.9 (2.25w)	S1.0 (2.5w)
3	Horizontal span (H)	:
		H1	H2	H3
4	Vertical span (V)	:
		V1	V2	V3

Based on the above input parameters, full-factorial 3⁴ = 81 samples have been designed. Among the four parameters, grid spacing has only a scaling effect, while the remaining have an effect on the shape of the sample as well. Hence, for ease of representation, the samples have been displayed as 27 samples each for S0.8, S0.9, and S1.0. The designs, as well as the tool paths of the same, were generated using MATLAB programming. As the shapes are along the build direction, the same paths are applicable for all layers. The tool paths were generated in a G-code format suitable for fabrication on a RepRap 3D printer, as described in the ensuing part in the Experimental methodology section.

Experimental methodology

All the designed samples have been fabricated through the material extrusion method using an acrylonitrile–butadiene–styrene (ABS) filament of 1.75 mm diameter. An open-source, RepRap-based, Protocentre 999 3D printing machine was used to fabricate the samples. The nozzle diameter of the setup is 0.4 mm. The temperatures of the nozzle head and bed platform were maintained at 230 and 100°C, respectively, throughout this study. Figure 4 illustrates various stages in fabrication of each of the samples; the samples thus fabricated for a given grid spacing are shown in Figure 5.

FIG. 4.

Various stages in creation of the 3rd-order Hilbert curve. Color images are available online.

FIG. 5.

Fabricated samples with various input variables for a given grid spacing. Color images are available online.

Compression experiments to test deflections for a constant load at a given point on the sample were performed with the help of a dynamometer mounted on a CNC machine. The sample was placed on the dynamometer and a ball-end tool of 5 mm diameter was clamped to the head of the CNC machine to exert the uniaxial load on the sample along the vertical direction (Z-axis), as shown in Figure 6. The bottom surface of the sample was kept fixed, while a point load was applied at the midpoint on the top surface by moving the ball-end tool slowly in the downward direction (i.e., negative Z-axis). To ensure that all deflections happen in the elastic range, the sampling load was applied to the specimen and a constant load of 2.5 N was selected.

FIG. 6.

Experimental setup for deflection test of Hilbert samples at various points. Color images are available online.

Results and discussion

The deflection results of all samples for a fixed load of 2.5 N are tabulated in Table 2. The same has been represented as a bar chart in Figure 7. These results have also been analyzed by using the analysis of variance (ANOVA) to identify influential parameters. The resulting plot for the mean deflection is shown in Figure 8.

Table 2.

Deflection of Samples (in mm) at a Load of 2.5 N

H	V	Annotation	S0.8			S0.9			S1.0
			O2	O3	O4	O2	O3	O4	O2	O3	O4
H1	V1	H1 × V1	0.74	0.61	0.26	0.71	0.74	0.35	1.02	1.03	0.57
H1	V2	H1 × V2	0.46	0.48	0.27	0.62	0.59	0.26	0.74	0.77	0.39
H1	V3	H1 × V3	0.40	0.36	0.21	0.41	0.46	0.23	0.75	0.71	0.34
H2	V1	H2 × V1	2.44	1.42	0.47	3.18	2.26	0.87	3.36	2.15	0.83
H2	V2	H2 × V2	2.00	1.58	0.45	3.20	2.36	0.73	3.33	2.25	0.36
H2	V3	H2 × V3	1.67	1.36	0.40	2.52	2.10	0.79	2.33	1.81	0.74
H3	V1	H3 × V1	3.00	1.52	0.58	3.99	2.45	0.90	4.87	2.96	1.28
H3	V2	H3 × V2	5.00	2.90	0.86	6.53	4.16	1.33	6.74	3.94	1.41
H3	V3	H3 × V3	4.43	1.34	0.73	6.85	4.72	1.32	5.56	3.67	1.00

FIG. 7.

Bar chart for deflection of samples by applying a load of 2.5 N on the central point. Color images are available online.

FIG. 8.

Mean effect plots for deflection using ANOVA. ANOVA, analysis of variance. Color images are available online.

It can be noted from Figures 7 and 8 that horizontal span (H) and order of the Hilbert curve (O) have a significant influence on the deflections. As the horizontal span increases, the distance between the supported ends also increases, thus leading to increased deflection. When the order of the Hilbert curve increases, it results in exponential increase in infill density of each cell, producing decreasing deflection for increasing order. As deflection is predominantly dependent on the distance between the supported ends and the infill density, vertical stacking (V) of these cells does not have much influence. The increase of grid spacing (S) increases the total size of each cell. Hence, it can be seen to have a small influence on the deflection. However, this influence is minuscule compared with those of horizontal span and order.

Mapping of the Control Variable with the Desired Parameter

Based on the experimental analysis, the order of the Hilbert curve and horizontal span were found to be the most influential parameters. In the creation of gradient structures, the process needs mapping of the desired variable (deflection in this case) to control variables. The use of multiple control variables will lead to multiple simultaneous solutions. Hence, to avoid this scenario, the horizontal span was picked over the order of the Hilbert curve as the preferred control variable. This selection has the following advantages: (1) compared with the order of the Hilbert curve, which is exponential in nature, horizontal span offers continuous change; (2) from the ease of fabrication, the infill density and spacing will remain constant across the component, the build time will also vary linearly with the volume of the part; and (3) as will be demonstrated in the subsequent analysis, horizontal span exhibits a continuously changing relationship with deflection.

Using the horizontal span as the control variable, an additional set of samples was fabricated to study the deflection behavior of samples at different loading locations of the sample component. The horizontal span H was varied from 1 to 7. The vertical span was kept at the minimum, that is, V1, as the number of vertical cells is seen to have no influence on the deflection. Similarly, grid spacing was also kept at minimum, S0.8, to allow better resolution in built samples. The order of the Hilbert curve was kept at the average value of O3. These fabricated samples have been tested for their deflection by applying a constant load of 4.0 N at various locations on the top surface of the samples, as shown in Figure 9. A constant load has been applied at the midpoint of each cell by moving the ball-end tool in the vertically downward direction, and the corresponding deflections have been listed in Table 3. These measurements provide the schema for selection of an appropriate H value for a desired deflection.

FIG. 9.

Experimental setup for deflection test of Hilbert samples at various points. Color images are available online.

Table 3.

Experimental and Simulation Results of Deflections of Samples at 4 N

Sample name	Distance from the midpoint	Numerical deflection (mm)	Experimental deflection (mm)	% Difference
H1	0.00	0.81	0.86	5.35
H2	−0.50	1.40	1.32	−6.06
	0.00	1.70	1.76	3.41
	0.50	1.40	1.28	−9.37
H3	−1.00	1.39	1.33	−4.51
	0.00	2.09	2.00	−4.50
	1.00	1.39	1.45	4.14
H4	−1.50	1.37	1.22	−12.30
	−0.50	2.10	1.70	−23.53
	0.00	1.92	1.88	−2.13
	0.50	2.10	1.90	−10.53
	1.50	1.37	1.35	−1.48
H5	−2.00	1.33	1.26	−5.98
	−1.00	2.09	1.98	−5.82
	0.00	2.10	2.00	−5.00
	1.00	2.09	1.87	−11.76
	2.00	1.33	1.20	−10.83
H6	−2.50	1.38	1.32	−4.55
	−1.50	2.10	2.04	−2.77
	−0.50	2.11	2.21	4.52
	0.00	1.92	2.14	10.28
	0.50	2.11	2.24	5.66
	1.50	2.10	2.14	2.02
	2.50	1.38	1.32	−4.28
H7	−3.00	1.36	1.33	−2.64
	−2.00	2.11	2.10	−0.48
	−1.00	2.10	2.14	1.87
	0.00	2.11	2.18	3.32
	1.00	2.10	2.16	2.55
	2.00	2.11	2.10	−0.36
	3.00	1.36	1.32	−2.84

To ensure the reliability of this mapping, these values have been corroborated with the help of numerical analysis. The numerical analysis has been performed using finite element solver on the ABAQUS 2018 to study the deflection behavior of samples at different loading conditions. The samples have been designed using a beam element of the rectangular profile of 0.4-mm thickness and 6.0-mm width. The mesh size had been kept as 0.05 mm. The mechanical properties of ABS polymer used in the analysis are density = 1.1 g/cc; Young's modulus = 1965 MPa; and Poisson's ratio = 0.38. The samples have been fixed at the bottom surface and a constant load of 4.0 N has been applied at the central point of the top surface of each unit cell in the vertically downward direction. The location of the loading point has been considered as zero at the midpoint of the sample and increased with unit cell distance in the left and right sides, as marked in Figure 10a, and the fabricated sample is shown in Figure 10b. Figure 10c shows the deflection result at the central point of samples (i.e., at zero position) in the Z direction for H6.

FIG. 10.

(a) Designed, (b) fabricated, and (c) finite element analyzed samples for H6. Color images are available online.

The deflections corresponding to the loading positions at the midpoint of each cell obtained from experimental and numerical methods have been mentioned in Table 3; the same has been plotted in Figure 11. Based on this, the experimental and numerical values can be said to be in good agreement, thus validating the approach. A graphical representation of deflection at the midpoint of samples versus the number of unit cells in the horizontal direction (i.e., deflection of samples at zero position, as listed in Table 3) is plotted in Figure 12. Here, deflection increases as the number of unit cells increases in the horizontal direction and saturates at a higher number of cells.

FIG. 11.

Deflection of Hilbert samples at various points for a constant load. Color images are available online.

FIG. 12.

Deflection at the midpoint of Hilbert samples versus the number of unit cells for a load of 4 N. Color images are available online.

Illustrative Case Study

Based on the above analysis, a case study of gradient shoe sole has been selected to demonstrate the effectiveness of the proposed method in achieving the gradient property. The foot soles of humans have different traits and their structures vary from one person to another. Hence, the pressure generated on different areas of the foot sole during walking appears gradient in nature due to their irregular shapes. To avoid these gradient pressures, the deflection of the shoe sole should have a gradient so that the pressure will be uniformly distributed on the entire foot sole. Therefore, a hypothetical shoe sole based on the gradient pressure map available in open literature was created for such an application.²⁷ The different color mapping of the gradient shoe sole, such as red, blue, and green colors, represents high, intermediate, and low pressure, respectively, as shown in Figure 13a. This image has been discretized into 14 × 17 pixels, with each pixel representing a unit Hilbert cell. This discretization is based on the minimum possible size of the cell (12.8 mm in the current scenario based on the grid spacing of S0.8). This design domain was further divided into 14 slices along the width of the shoe sole, as shown in Figure 13b. Each slice has been generated using Hilbert curves with the horizontal span dependent on pressure values; the high-pressure zone will have a higher horizontal span (H) and the low-pressure zone will have a lower H. An illustration of such deflection and horizontal span mapping is shown in Figure 14. In this figure, the bar chart indicates displacement required at various locations of the pixel array. These are mapped to the pressure response values plotted in Figure 11 and indicated by dots and corresponding line curves in the figure. The stacked Hilbert cells thus obtained are shown on the top. Such arrangement of Hilbert cells for all 14 slices is shown in Figure 13c. All these 14 slices have been fabricated separately using the same machine parameters as employed earlier and then stacked/glued together on a plate, as shown in Figure 13d.

FIG. 13.

Various stages in fabrication in the illustrative case study: (a) pressure distribution on foot sole, (b) discretized pressure data, (c) Hilbert curves for each slice of the geometry, and (d) fabricated slices stacked together. Color images are available online.

FIG. 14.

Mapping of deflection and H spacing: bar chart indicates displacement required; dots and corresponding line curves indicate pressure response values; the stacked Hilbert cells thus obtained are shown on the top. Color images are available online.

The sole model thus fabricated was tested for deflection by applying a uniform load on the top surfaces. A constant load of 8 N has been applied on the top surface at the center of each unit cell of the shoe sample. The corresponding deflection in the negative Z direction (i.e., in the vertically downward direction) has been measured under the elastic condition to analyze the gradient deflection behavior. The deflections of the fabricated sample were measured at different locations by applying the uniform load to analyze their gradient deflection behavior. The experimental result of deflections at various points of the sample in the Z direction (i.e., in the vertically downward direction) is plotted in the form of a color map in Figure 15, with the measured deflection values indicated for each pixel. The low-pressure regions of the sample have less deflection, varying in the range of 0.72–1.12 mm. Similarly, intermediate- and high-pressure regions have deflection ranges nearing 1.12–1.92 and 1.92–2.33 mm, respectively. As can be observed by comparing the intended pressure map of Figure 13b and resultant deflection map of Figure 15, the measured deflections match the desired foot pressure variability, thus validating the proposed approach to achieve gradient properties.

FIG. 15.

Color contour map of measured deflection of the fabricated foot sole. Color images are available online.

Conclusions

A methodology to fabricate components with tailored properties for material extrusion-based processes has been presented. Following a brief discussion on various infill patterns for tailoring the properties in a localized manner, Hilbert infill patterns have been chosen. The Hilbert curve has a periodic and continuous infill path, making it easy to achieve the gradient property by controlling input parameters during design and fabrication. Of the two possible loading directions (defined with respect to build direction), the parallel scenario has a continuous Hilbert curve, but suffers discontinuity of material for loading. Hence, perpendicular loading is chosen here for achieving gradient compliance behavior.

Four input variables, order of the Hilbert curve (O), grid spacing (S), and horizontal (H) and vertical (V) spans, have been analyzed to study their effects on deflection behavior. Among these, O and H were found to have considerable influence on deflection, while S and V showed an insignificant effect. Based on this, a mapping schema for selecting the required control parameter values for a desired amount of deflection has been worked and corroborated with numerical simulation in ABAQUS.

Subsequently, a case study of gradient pressure on the shoe sole was taken up as a demonstration exercise. The varying pressure generated at different areas of the foot was mapped to deflection, which in turn was mapped to varying H values for each cell of the discretized domain. The fabricated sample was tested to obtain deflections for each cell at a constant load and compared with gradient pressure data. The behavior of the fabricated sample was in sync with the desired foot pressure data, validating the methodology.

The current study gives the basic framework for generating the gradient density component, which can be used for various applications such as biomedical, prosthetic, and orthopedic applications. In the future, the study can be extended to generate gradient samples in three dimensions of the components using 3D Hilbert curves.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

No funding was received for this article.