## 3 Modeling of Cellular Lattice Structure  
As discussed, use of \(R\) functions are more efficient than boundary representation and voxel based modeling techniques. We propose to construct a series of objects having unique cellular lattice structure inside and within the closed boundaries, using trigonometric functions and \(R\)-functions. For working with function representation, \"HyperFun, a modeling language\", a simple specialized high level programming language and associated software, designed to set out function based models, is used [50]. To construct the series of cellular lattice structure within a closed cylinder, an algorithm is developed and given at the end of the section. The series is constructed by assigning values to variables, \(frequency\)\((f)\) and \(vertical\)\(shift\)\((v)\) used in sin function. The values of frequency are taken, which are 19 (\(f_{1}\)) and 21 (\(f_{2}\)), whereas the vertical shift is varied from \"0\" to \"0.4\" with an increment of \"0.1\". Frequency controls the length of the period, and vertical shift controls the division within a period. The selected values of frequency and vertical shift are according to the size limitations of test specimen following the standard ASTMD1621-16 and printing capabilities of 3D printing machine. The internal lattice structure consists of intersecting vertical and horizontal hollow bars, generated through function representation, as shown in Figure 4a. Figure 4b provides the x-sectional view of the specimen through the bars to show the hollow regions of the bar. During printing, material accumulates to form the outer surfaces of the hollow bar, and the same material (PLA) accumulates in the space between the hollow bars to form intersecting vertical and horizontal column-like structure (solid bar). Some extra material is  
Figure 4: X-sectional views of test specimen with (**a**) complete lattice structure within the shell (**b**) internal structure of the hollow bar. Figure 3: Dimensions of test specimen (**a**) Test specimen overall dimensions (**b**) X-sectional view of test specimen with internal cellular lattice structure (**c**) X-sectional view of hollow test specimen (solid part). consumed to act as support material and provide support to hollow bars. Figures 5 and 6 show the internal structure of a test specimen after rendering plus slicing and printing operations. The variation in the thickness of the hollow bars and material distribution for the same frequency and different values of vertical shift can be seen by comparing Figure 5 with Figures 6a and 5b with Figure 6b, respectively. The outer shape, inside structure, and geometry of the rendered and printed test specimen is shown in Figure 7a-c, respectively. Figure 7b shows the magnified view of the internal structure containing the dimensions after rendering. Here, \"p\" is the thickness of the hollow bar and \"q\" is the distance between the hollow bars, which is empty space. The empty space between the hollow bars, shown in Figure 7b, is then filled by the same material (PLA) during printing, as shown by Figure 7c, to form solid bars. Tables 3 and 4 show dimensions \"p\" and \"q\" in millimeters related to each specific test specimen. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Frequency (\(f\))**}} & \multicolumn{5}{c}{**Vertical Shift (\(v\))**} \\  & **0** & **0.1** & **0.2** & **0.3** & **0.4** \\ \hline \(f_{1}\) & 19 & 4.2 & 4.5 & 4.8 & 5.3 & 5.3 \\ \(f_{2}\) & 21 & 3.8 & 4.3 & 4.5 & 4.9 & 4.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Thickness of hollow bars (non-filled sections) in millimeters (p). Figure 5: X-sectional top views of test specimen with frequency \"19\"and vertical shift “0” (**a**) after rendering plus slicing (**b**) after printing. Figure 6: X-sectional top views of test specimen with frequency “19” and vertical shift “0.4” (**a**) after rendering plus slicing (**b**) after printing.