# A Numerical Approach for the Efficient Concept Design of Laser-Based Hybrid Joints
## 3 Demonstration of the Approach
### Optimization Setup  
The previously introduced models for concept development, strength calculation, and cost calculation were combined in an optimization procedure as shown in Figure 11. The execution of all models is fully automated and the parameters are interconnected. Typically, there are several possible solutions for a given problem. The task of the optimization algorithm is to identify one satisfying solution for the design variables which fulfills the requirements. The strength and costs were defined as objectives. For the strength, the following requirement was defined: the strength must be at least equal to the benchmark and the costs shall be simultaneously minimized. Multiple classical state-of-the-art concepts (linear, parallel cavities) were used with the distances of 300 \(\upmu\)m, 400 \(\upmu\)m, 500 \(\upmu\)m, and 600 \(\upmu\)m as benchmarks. The strength and cost values of the benchmarks (production of one component) are shown in Table 2. As expected, costs and strength decreased as the distance between structures increased. Additionally, the number of cavities and the mark distance are given. This information is useful for the evaluation of the laser production effort and the induced thermal stress. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Type** & **Distance (\(\upmu\)m)** & **Costs (\(\ell\))** & **Max. Load (N)** & **Cavities** &
\begin{tabular}{c} **Mark Distance** \\ **(mm)** \\ \end{tabular} \\ \hline a (classical) & 300 & 0.00237 & 2831 & 40 & 5904 \\ b (classical) & 400 & 0.00178 & 2475 & 30 & 4428 \\ c (classical) & 500 & 0.00148 & 2363 & 25 & 3690 \\ d (classical) & 600 & 0.00119 & 2021 & 20 & 2952 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Benchmark values for the classical design with linear and parallel structures. Figure 10: Resulting concepts for the demonstration example. In (**a**) and (**b**) the theta and phi angles are shown. In (**e**) the von Mises distribution is shown. (**f**) shows the application of the stressThreshold for a value of 0.1. In (**c,d,g**) the use of two, three and four clusters are shown. With the given strength constraint, the costs shall then be minimized. The parameters stressThreshold and the distance for each cluster were used as design variables. Theoretically, the microstructure of each cluster (i.e., the geometry of the cavities) could also be varied through the number of laser runs. However, only the microstructure with six laser runs was used to limit the number of design variables and the initial optimization effort. This microstructure has distinctive undercuts and high strength as shown in ref. [18]. Choosing an appropriate optimization algorithm is a challenge. Typically, a compromise must be found between fast convergence and the wide coverage of the solution space. If convergence is too fast, there is a risk of becoming stuck in local minima too quickly. With algorithms that are too free, the solution space can be covered widely and the chances increase to identify a global optimum. However, convergence can sometimes not be achieved and the optimization process requires numerous iterations. Various optimization algorithms were tested in pre-investigations (e.g., PSO, NSGA-II, gradient-based algorithms, etc.). The other algorithms tested required too much time for convergence, so efficiency of the whole approach was reduced. However, the final results did not differ from those obtained with NSGA-II. Especially the gradient-based algorithms converged very quickly. As a result, the outcomes were not satisfactory. Genetic optimization with NSGA-II was chosen as the best optimization algorithm for the given problem because it is a commonly applied robust algorithm and can handle multiple objectives [44; 45]. The multi-objective optimization gives multiple Pareto-optimal solutions as a result. From these different solutions, the engineer can finally choose their preferred solution. For future extensions of the approach, multi-objective optimization can also be easily extended with additional criteria.