# Optimized One-Click Development for Topology- Optimized Structures +
## 2 Materials and Methods
### Topology Optimization for Stiffness and Strength  
For TO, the freeware Z88 Arion(r) was used, which was based on density methods for TO. While in general, the SIMP and rational approximation of material properties (RAMP) were implemented, SIMP was used for all optimizations presented in this article. The SIMP method thereby is given as  
\[E(\rho_{i})\ =\rho_{i}{}^{p}E_{0}, \tag{1}\]  
where \(E\) describes the adapted Young's modulus of the material, respectively, \(E_{0}\) describing the Young's modulus for the solid material, \(\rho\) is the design variable and \(p\) the penalization factor [17,40]. The used TOSS algorithm was developed in [40]. It is a hybrid TO algorithm, where first a minimum compliance optimization problem is solved by using the optimality criterion method (OC) [17]. The optimization problem for this first step can be written as follows:  
\[\text{min}C\ =\text{min}u^{T}Ku, \tag{2}\]  
where \(C\) denotes the compliance, \(u\) describes the displacement vector and \(K\) the stiffness matrix. The optimality criterion \(B_{i}\) after [18,40] is then given as  
\[B_{i}=\ \frac{u^{T}\frac{\partial K}{\partial\rho_{i}}u}{\partial v_{i}^{0}}\ =1+\ \frac{\lambda_{i}}{\partial v_{i}^{0}}-\ \frac{\mu_{i}}{\partial v_{i}^{0}}, \tag{3}\]  
with \(\theta\), \(\lambda\) and \(\mu\) as Lagrange multiplicators and \(v_{i}^{0}\) the volume of the initial elements. To satisfy the optimality criteria in Equation (3), \(B_{i}\) has to be equal to 1 for all elements \(i\). This finally leads to the following iteration rule:  
\[\rho_{i}^{k+1}=\ \begin{cases}\max\Big{(}0,\ \rho_{i}^{k}-\alpha\Big{)}&if\ \rho_{i}^{k}B_{i}^{\eta}\ \leq\max\Big{(}0,\ \rho_{i}^{k}-\alpha\Big{)}\\ \rho_{i}^{k}B_{i}^{\eta}&if\ \max\Big{(}0,\ \rho_{i}^{k}-\alpha\Big{)}\ <\ \rho_{i}^{k}B_{i}^{\eta}\ <\min \Big{(}1,\ \rho_{i}^{k}+\alpha\Big{)}\\ \min\Big{(}1,\ \rho_{i}^{k}+\alpha\Big{)}&if\ \rho_{i}^{k}B_{i}^{\eta}\ \geq\min \Big{(}1,\ \rho_{i}^{k}+\alpha\Big{)}\end{cases}  \tag{4}\]  
with a step length \(\alpha\) and a damping factor \(\eta\) so that a decision for the new design variable can be made with consideration of the design variable at iteration step \(k\). After convergence was obtained, a strength optimization was started based on empirical TO approach. In this case, the soft kill option (SKO) was used, which used the biological rule of growth to maximize the strength. This was done by achieving a homogeneous surface tension [18,40,44]. In this procedure, during the finite element analysis (FEA) for each node \(j\) the equivalent stress \(\sigma_{i}^{k-1}\) was computed. The equivalent stress was then used to simulate the biological rule of growth. In contrast to mathematical TO approaches, [PERSON]'s modulus of the elements was directly modified. This modification can be easily done by varying the node temperatures of the finite elements. It emphasizes here that these temperatures have no physical meaning and are only used to easily modify [PERSON]'s modulus. This possible since [PERSON]'s modulus can be defined as a function of the element temperatures in every FEA software [18]. Therefore, a virtual temperature \(T^{k}\) was computed for each node \(j\) inside the design space:  
\[T^{k}_{j}\ =\ \hat{T}^{k}_{j}+s\Big{(}\sigma^{k-1}_{i}-\ \sigma_{\text{ref}} \Big{)}, \tag{5}\]  
with:  
\[\hat{T}^{k}_{j}\ =\ \left\{ \begin{array}{cc}100&if\ \hat{T}^{k-1}_{j}\ >100\\ \hat{T}^{k-1}_{j}&\text{else}\\ 0.1&if\ \hat{T}^{k-1}_{j}\ <0.1\end{array} \right. \tag{6}\]  
and the scaling factor \(s\) and reference stress \(\sigma_{\text{ref}}\), which are the user-defined inputs. The virtual temperature is then calculated for each element \(i\):  
\[T^{k}_{i}\ =\ \frac{1}{n_{E}}\sum_{j=\ n_{f}}^{n_{i}}\ T^{k}_{j} \tag{7}\]  
where \(n_{E}\) represents the number of nodes per element, correspondingly describes \(n_{f}\), the first node which belongs to the element and \(n_{I}\), the last node.