# A Numerical Approach for the Efficient Concept Design of Laser-Based Hybrid Joints
## 2 Method and Models
### Cost Model  
The total costs of a product are typically made up of various components such as manufacturing costs, material costs, and overhead, etc. In this paper, only the influence of the microstructure on the strength was considered. Plastic manufacturing costs do not change for different microstructures if the dimensions of the joining zone remain the same. Therefore, only the costs of laser manufacturing were considered, which have a direct effect on joint strength. Other costs such as overhead costs vary from company to company and must therefore always be calculated separately for each case. For the calculation of laser costs, the calculation of a machine hourly rate \(\mathrm{C_{M}}\) and the laser production time \(\mathrm{T_{L}}\) are the most important factors (see Equation (1)). The only approach to calculate laser production time and costs so far can be found in ref. [16]. In this approach, the laser production time is calculated using the length \(\mathrm{l_{i}}\), distance \(\mathrm{d_{i}}\), and number of runs for each cavity together with the scanning speed vs. of the laser head (see Equation (2) and Figure 8).  
Figure 8: Arbitrarily shaped surface with dimensions of the individual cavities.  
Figure 7: (**a**) Bilinear traction-separation law; (**b**) multidimensional traction-separation law with trapezoidal damage evolution.  
The structures themselves are processed at the desired scanning speed. However, the high speed of \(\sim\)10 m/s results in substantial influences due to the mass inertia of the galvo scanner system [42]. In order to avoid burn-in and to manufacture clean and reproducible cavities, the laser-head performs a compensation movement (so-called skywriting) between two structures, which leads to an additional delay. This effect is considered in the factor T\({}_{\text{D}}\) and can be estimated as described in ref. [43]. Due to the skywriting effect, many short cavities take longer for production than few long cavities.  
\[\text{C}_{\text{L}} =\text{T}_{\text{L}}\text{-C}_{\text{M}} \tag{1}\] \[\text{T}_{\text{L}} =\frac{\sum_{i=1}^{n}\text{w}\cdot\text{N}_{i}\text{+d}}{\text{v }_{\text{s}}}\text{+T}_{\text{D}}\] (2) \[\text{C}_{\text{M}} =\frac{\frac{\text{C}_{\text{L}}}{\text{tw}}\cdot(1+\text{C}_{ \text{m}})\text{+F}\cdot\text{C}_{\text{F}}}{\text{t}_{\text{r}}}\text{+C}_{ \text{E}} \tag{3}\]  
The machine hourly rate C\({}_{\text{M}}\) was calculated using the capital costs, write-off time, and relative maintenance costs C\({}_{\text{m}}\). Furthermore, the building costs were considered by the footprint F and the building rent costs C\({}_{\text{F}}\) (see Equation (3)). These values were divided by the maximum running time of the machine t\({}_{\text{r}}\). In addition, the energy costs C\({}_{\text{E}}\) were taken into account. Personnel costs were not included as they cannot be estimated yet for an individual component. The specific values for the parameters are given in Table 1. Since laser machines are still usually prototype systems, these values are based on empirical knowledge. By multiplying the machine hourly rate by the laser production time, the laser costs could then be calculated for any given microstructure concept (see Equation (1)).