## 5 Applying Volume to the Fractal Structures
### The CAD Method
## 6 Case Studies

### Creating Fractal Structures on a Surface Mesh Using the Geometric Method

For the first case study, the geometric computational method was used. The cad shell was defined using a surface mesh that had two components; the first component (the base of the shell) was defined by a planar structure (parallel to the _XOZ_ plane) with 5 mm thickness and the second component (the top of the shell) was defined by a curved surface with thickness of 5 mm and the following equation:

\[y=x\cdot\exp\left(-x^{2}-z^{2}\right), \tag{21}\]

which was scaled by a factor of 30 in the \(X\) and \(Z\) directions and by a factor of 25 in the \(Y\) direction. Furthermore, the surface was translated by 55 mm in the \(y\) direction. Four fractal structures were defined with different origins, branch angles, and branch lengths. The maximum allowed depth for each fractal in all four cases was \(d=5\), and the radii for all fractals were \(r=1.5\) mm. For the fractal origins, the following parameters were used:

\[ \begin{split} ORG_{1}&:[V_{X1},V_{Y1},V_{Z1}]^{ \mathrm{T}}=[10,1,0]^{\mathrm{T}},\theta_{1}=0,\theta_{2}=0,\theta_{3}=0,\\ ORG_{2}&:[V_{X2},V_{Y2},V_{Z2}]^{\mathrm{T}}=[20,1,20]^{\mathrm{T}},\theta_{1}=0,\theta_{2}=0,\theta_{3}=0,\\ ORG_{3}&:[V_{X3},V_{Y3},V_{Y3}]^{\mathrm{T}}=[-25,1,10]^{\mathrm{T}},\theta_{1}=0,\theta_{2}=0,\theta_{3}=0,\\ ORG_{4}&:[V_{X4},V_{Y4},V_{Z4}]^{\mathrm{T}}=[-10,1,-10]^{\mathrm{T}},\theta_{1}=0,\theta_{2}=0,\theta_{3}=0.\end{split}  \tag{22}\]

Furthermore, Tables 11-14 in Appendix A show the input geometric data used to define the four tree-like fractals (the tables were constructed accruing to Tables 1 and 2). As a reminder, each row in the tables represents the depth of the tree-like fractals, whereas each column represents the parameters needed to compute each node. The intersection of the fractals with the surface mesh was computed, resulting in a new set of data tables (Tables 11-14 in Appendix A) for the trimmed tree-like fractals. Figure 10 shows the computed fractals (after applying volume and uniting them with the shell according to the steps described in Section 5.1). To export the surface mesh into an STL file format, the function \"stlwrite\" [22] from Matlab was used, which takes as inputs the cartesian points of the surface and the triangulations (both being computed in the algorithm described in Section 5.1). At this point, step#4 is completed from the general algorithm (see Section 2), and the part is ready for 3D printing or further CAD tests such as FEA (th \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Method**} & **Works with** & **Works with** & **Prone to** & **Requires** & **Certified** & **Relative** \\  & **Input Surface** & **Input CAD** & **Computational** & **Fractal** & **Result** & **Computation** \\  & **Mesh** & **Model** & **Errors** & **Trimming** & **Time** \\ \hline Geometric & & & Small & & & \\ method & Yes & no & probability & Yes & No & Medium \\ \hline \multirow{2}{*}{CAD method} & \multirow{2}{*}{Yes} & \multirow{2}{*}{Yes} & Very small & & & \\  & & & probability & No & SCAD & Low \\ \hline \end{tabular}
\end{table} 
Table 6: Broad comparison between the geometric and the CAD methods. ### Creating Fractal Structures on a Hemispherical Shell Using the CAD Method

For the second case study, a hemispherical CAD shell was chosen as an input. This has two components: the first component is a concave hemisphere (radius of 50 mm and thickness of 5 mm) and the second is a convex hemisphere (radius of 5 mm) with its center displaced by 5 mm (with respect to the concave hemisphere) on the \(z\)-axis. The maximum allowed depth of the fractals was \(d=5\), and the radii for all fractals were \(r=1\) mm. Twenty-four fractal structures were generated using the following relation:

\[ \begin{split} OR_{jk}:\left[V_{Xjk},V_{Yjk},V_{Zjk}\right]^{ \mathrm{T}}&=[0,0,5]^{\mathrm{T}},\theta_{1}=j\frac{\pi}{3}+k \frac{\pi}{6},\theta_{2}=0,\theta_{3}=k\frac{\pi}{6},\\ j&=1\dots 6,k=1\dots 4.\end{split}