# A Numerical Approach for the Efficient Concept Design of Laser-Based Hybrid Joints
## 2 Method and Models
### Automated Generation of Microstructure Concepts
Therefore, isotropic properties with a low Youngs modulus were applied. Using an isotropic material behavior for the interlayer is important to ensure that the directions of the stress vectors are not pre-affected by anisotropic material behavior [31]. Preliminary investigations showed good results for a Young's modulus of 100 MPa. Figure 3: Mechanical characteristics of laser-based plastic–metal joints based on a FE-Model of a single cavity (RVE). This interlayer obtained the real mechanical properties of the microstructure in later steps, after the microstructure is defined. After the application of the load case in the FE-model, the stress tensor was extracted from the mesh elements of the interlayer. From the stress tensor, the magnitude of the stress (von Mises) and the orientation of the stress vector could be calculated for each individual mesh element. Typically, the stress varies across the surface of the components. Some areas are highly stressed and some are low stressed or even not stressed at all [32]. For cost reasons, very low-stressed areas can be excluded from further analysis in order to save production costs and reduce thermal effects during laser production. Thus, the elements were filtered based on von Mises stresses using the parameter stressThreshold. Elements with a stress \(\sigma_{\text{element}}<\text{ stressThreshold}\cdot\sigma_{\text{max}}\) were filtered and are not further considered. The idea of using a stressThreshold to filter low-stressed areas was successfully demonstrated in ref. [32]. Figure 4 shows the application of the stressThreshold for values of 0, 0.5, and 0.75. Figure 4 shows a generic stress distribution. The stress was the highest at the corners and decreases towards the center. It is apparent that the high stresses covered only a small fraction of the area. The application of the stress limit value of 0.5 (elements with stress above 50% of the maximum stress) resulted in filtering out a large area (gray areas). Using a stress threshold of 0.75 resulted in even more elements being filtered out. The specific stressThreshold value cannot be predicted upfront and needs to be iteratively determined. How to optimally select the stressThreshold value is determined later in the optimization approach (see Section 3). The stress vector is decomposed into an in-plane angle (phi) and an out-of-plane angle (theta) since these are the relevant directions of anisotropy (see Figure 5). Besides the von Mises stress, the theta and phi angles also vary across the joining zone. Each element would have to be structured individually to realize the best cost- and load-optimized concepts. However, as mentioned above, the cavities must always be continuous lines and punctual structuring is not possible nor economically feasible. Thus, the main challenge is to identify connected regions that have similar properties in terms of von Mises stress and phi and theta angles. Figure 4: The stress threshold is used to reduce the microstructured area by filtering low-stressed elements. To solve this task, a cluster algorithm was used which groups the elements into a finite number of contiguous areas with similar properties (von Mises, phi, and theta). For each of these areas an appropriate microstructure can be assigned. The actual assignment of the optimal microstructure for given cost and strength requirements was solved using an optimization algorithm (see Section 3). As input for the cluster algorithm, the element-wise von Mises stresses and theta and phi angles as well as the corresponding coordinates are needed, which are normalized beforehand to the range of \(0<\chi<1\). Without normalizing, the different value ranges would result in overweighting one factor and thus to a strong distortion and scattered clusters. There are numerous options for cluster algorithms which differ in their functionality, computation time, and results. An overview can be found in ref. [33].