## 5 Applying Volume to the Fractal Structures
### The CAD Method
 \tag{23}\]

Furthermore, a hemispheric surface mesh was provided, defined for the inner surface of the CAD shell using the equation:

\[x^{2}+y^{2}+z^{2}-45^{2}=0,z\geq 0 \tag{24}\]

If the CAD shell is based on a well-defined surface, it is convenient to compute a surface mesh to compute the trimmed fractals. Table 10 in Appendix B presents the input geometric data (branch angles and lengths) for the tree-like fractals used in this case study. Tables 11 and 12 show two examples of tables computed for the trimmed fractals (not all fractal data are reported as they are lengthy and not relevant to the results presented in this subsection). Figure 11 shows the final CAD part (computed using MatSCAD) with the two hemispheres and the fractal structures uniting them. ### Creating Test Samples with Fractal Structures Using the CAD Method

For the third case study, two test samples were designed using the CAD method to study how the fractals behave under mechanical stress. The first sample was designed for a compression test, and the shell contains two planar parts (square shapes of 20 mm side and 1 mm thickness, displaced by 20 mm) that are united through six fractal structures. The origins of the fractals are defined by:

\[ \begin{split} ORG_{1,2}:[V_{X1,2},V_{Y1,2},V_{Z1,2}]^{\text{T}}=[ \pm 5,-5,0]^{\text{T}},\theta_{1}=0,\theta_{2}=0,\theta_{3}=\frac{\pi}{2},\\ ORG_{3,4}:[V_{X3,4},V_{Y3,4},V_{Z3,4}]^{\text{T}}=[\pm 5,0,21]^{\text{T}}, \theta_{1}=0,\theta_{2}=\pi,\theta_{3}=\frac{\pi}{2},\\ ORG_{5,6}:[V_{X5,6},V_{Y5,6},V_{Z5,6}]^{\text{T}}=[\pm 5,5,0]^{ \text{T}},\theta_{1}=0,\theta_{2}=0,\theta_{3}=\frac{\pi}{2}.\end{split}  \tag{25}\]

The radii of all fractal structure branches were 1 mm. Table 12 in Appendix C shows the input geometric data for the fractal structures, and Tables 13 and 14 show the branch data for the fractals. Figure 12 shows the CAD model of the compression sample. The second test sample was designed for the bending tests. Again, the sample shell contains two planar components (rectangular shapes with sides 10 mm and 80 mm, respec

Figure 11: Final CAD part with two hemispheres united by fractal structures: (**a**) bottom view (plane _XOY_) and (**b**) isometric view. Figure 12: CAD sample part for compression tests: (**a**) front view (plane _XOZ_); (**b**) side view (plane _YOZ_); and (**c**) isometric view. tively, with 1 mm thickness and displaced by 9.5 mm). The fractal structures were defined with the following origins:

\[ \begin{split} ORG_{1,i}:[V_{X1,i},V_{Y1,i},V_{Z1,i}]^{\text{T}}& =[i\cdot 5,-5,0]^{\text{T}},\theta_{1}=0,\theta_{2}=0,\theta_{3}= \frac{\pi}{2},\\ ORG_{2,j}:[V_{X2,j},V_{Y2,j},V_{Z2,j}]^{\text{T}}&=[ j\cdot 5,0,21]^{\text{T}},\theta_{1}=0,\theta_{2}=\pi,\theta_{3}= \frac{\pi}{2},\\ ORG_{3,k}:[V_{X3,k},V_{Y3,k},V_{Z3,k}]^{\text{T}}&=[ k\cdot 5,5,0]^{\text{T}},\theta_{1}=0,\theta_{2}=0,\theta_{3}=\frac{\pi}{2},\\ \end{split}  \tag{26}\]

\[ \begin{split} i,j,k&=-9...9.\end{split} \]

The radii for all of the branches were set to 1 mm. Table 15 from Appendix D shows the geometric input data for the fractals, whereas Tables 16 and 17 show the branch data for the fractals. Lastly, Figure 13 illustrates the test sample for bending in CAD. Exporting the test parts to an STL file format concludes step#4. ## 7 Conclusions

Internal structures, such as lattice structures are of great interest for various domains of AM, and a lot of focus is given worldwide to understanding their properties. Tree-like fractals can also be used as internal structures in AM, but no significant progress has been reported in their modeling and implementation for AM. This paper presented a computational model designed to generate tree-like fractals as internal structures in AM mechanical parts. The computational model brings several contributions with respect to the use of tree-like fractals in AM. Firstly, a comprehensive and detailed mathematical model is derived to construct various tree-like fractal structures. The fractals are viewed as mathematical structures with clearly defined properties.