## 2 Materials and Methods
### Finite Element Model and Optimization Setup
#### 2.3.2 Splitting and Linking of the Lattice  
After the FEA model is set up in HyperMesh, the lattice is split into ten separate, equidistant regions along its height, as marked by the colours in Figure 7. The region identifier numbering scheme starts from T1 at the top of the lattice to T10 at the bottom. Each region possesses a single thickness attribute, and every layer of the unit cells is split in half. The RADIOSS solver input file is then exported for the subsequent steps. Altair HyperStudy v.2021.2, a general optimization suite able to control and assess virtually any executable calculation software and its outputs [29], is used to link the separate lattice regions, control their thickness, and perform the optimizations. The procedure demonstrated here makes use of the fact that the thickness assignment of idealized 1D and 2D elements is achieved by providing a single thickness value in the solver input file. A section describing the beam element properties of a RADIOSS v.2021.2 input file is shown in Figure 8a. It contains information about its cross-section; iteration point distribution; and, among other things, the element group's radius assignment marked by a red rectangle. HyperStudy allows to parameterize it, as shown in Figure 8b, and it can then be altered freely for parameter studies or optimization runs. Figure 8: Beam element property definition of a RADIOSS model input file with its thickness property marked in red (**a**) and the same section after parameterization of the thickness value in HyperStudy (**b**). Figure 7: Splitting and linking of the lattice geometry. The graph on the right schematically shows the stepwise thickness distribution linked to mathematical functions that can be defined arbitrarily. The detail on the left side shows the subdivision of the beam into ten beam elements for increased accuracy. Upon importing the input file into HyperStudy, it is parameterized by defining each thickness property as an optimization variable. However, performing optimization runs with ten separate variables would require significant computational resources. Therefore, the thickness parameters are linked by a mathematical function using the lattice's height as an independent variable. The lattice's origin point is located at the top layer, while the height is assessed at the lower edge of each region (at 5 mm, 10 mm, etc. ), as indicated by the arrows in Figure 7. Three different functions are considered:  
\[\text{Linear law}:\;\text{T}(\text{h})=\text{a}*|\text{h}+\text{c}|+\text{b} \tag{1}\]  
\[\text{Quadratic law}:\;\text{T}(\text{h})=\text{a}*(\text{h}+\text{c})^{2}+ \text{b} \tag{2}\]  
\[\text{Power law}:\;\text{T}(\text{h})=\text{a}*\text{h}^{c}+\text{b} \tag{3}\]  
The linear law (Equation (1)) was chosen to add to the investigations reviewed in Section 1.2, which also considered linear thickness progressions and demonstrated significant performance improvements. It is defined such that it can include an inflexion point via the insertion of the absolute value function. The quadratic law (Equation (2)) follows a similar approach but is intended to provide larger thickness differences between regions than the linear law to investigate any potential differences. Both functions include a scaling factor, \(a\); a constant factor, \(b\); and a shifting factor, \(c\), allowing the inflexion point to be altered. The limits of the shifting parameter, \(c\), allow the inflexion point to be moved outside of the considered range, which also permits purely increasing or decreasing thickness progression. The power law (Equation (3)) makes identical use of the \(a\) and \(b\) parameters, but the \(c\) parameter acts as an exponent in this case. It is set up symmetrically with respect to the middle plane (i.e., T1 is equal to T10, T2 is equal to T9, etc.) to prevent extreme thickness values at the upper and lower regions. The linear and quadratic laws possess no such inherent symmetry.