# On the Development of a Surrogate Modelling Toolbox for Virtual Assembly
## 4 Use Case Study for a Laser Welding Assembly
### Validation against the Physical Assembly
#### 4.2.1 Comparison of Variation Distributions  
In order to quantify the variation, the mean variation \(\overline{\mathrm{d}}\) is introduced, which is defined as the arithmetic mean:  
\[\overline{\mathrm{d}}=1/\mathrm{W}\cdot\sum_{w=1}^{\mathrm{W}}||\mathrm{P_{w} }-\overline{\mathrm{P}}||_{2} \tag{4}\]  
To assess the mean variation of the virtual assemblies and the physical assemblies, \(\overline{\mathrm{d}}_{\mathrm{VA}}\) and \(\overline{\mathrm{d}}_{\mathrm{PA}}\) respectively, are considered in the following. In Equation (4), \(||\mathrm{P_{w}}-\overline{\mathrm{P}}||_{2}\) is the Euclidean distance between point \(\mathrm{P_{w}}\) and average point \(\overline{\mathrm{P}}\) from \(\mathrm{W}\) repetitions. In our work, 24 repeated virtual assemblies and three physical assemblies are studied. \(\overline{\mathrm{P}}\) is determined as the mean coordinate from the repeated scans, where points are associated by means of a k-nearest-neighbor search. The quantities \(\overline{\mathrm{d}}\) and \(\mathrm{P}\) are computed for all \(\mathrm{N}\) point associations \(\mathrm{P_{w,n}}\); \(\overline{\mathrm{P}}_{n}\), where \(\mathrm{n}\in[1;\mathrm{N}]\) is the point index variable.  
The average manufacturing deviations \(\mathrm{d_{C}}\) and \(\mathrm{d_{H}}\) of the covers and housings, respectively, are determined analogously to Equation (4) as mean deviations from the respective mean value for all 24 covers and housings. They are normally distributed (their mean values are equal to approx. 0.03 mm) and are about ten times smaller than the mean variation of all physical assemblies \(\overline{\mathrm{d}}_{\mathrm{PA}}\) (up to max. 0.4 mm). Values for \(\overline{\mathrm{d}}_{\mathrm{PA}}\) are shown in Figure 8 (right). Thus in a first approximation, the influence of the manufacturing deviations on assembly variations is neglected, because they contribute to less than 10% of the extent of the variation of all physical assemblies \(\overline{\mathrm{d}}_{\mathrm{PA}}\).  
The propagated uncertainty \(\mathrm{u_{P}}\) of the virtual assembly comprises the uncertainty of measurement points \(\mathrm{u}\) that arises from 20 CT scan repetitions. The uncertainty up is further increased by the transformation uncertainty emerging at VA, which causes a slight variation of the determined assembly position and orientation of the point set to register for repeated registrations. The subscript \"P\" denotes the transformed state that is gathered by using a linear uncertainty propagation method. Here, an uncertainty coverage factor of \(\mathrm{k}=1\) as defined in ISO/IEC 98-3 [69] is applied. The uncertainty \(\mathrm{u_{P}}\) is shown in Figure 8  
Figure 8: Propagated uncertainty \(\mathrm{u_{P}}\) at the cover (**left**). Mean variation \(\overline{\mathrm{d}}_{\mathrm{PA}}\) of all physical assemblies (**right**).  
(left) and is in range of up to ca. 0.07 mm. For the transformed point set, up equals to the standard deviation of the VA variation \(\overline{\mathrm{d}}_{\mathrm{VA}}\).  
Figure 9 shows the mentioned distributions of assembly and manufacturing deviations. The bias \(\overline{\mathrm{B}}\) represents the mean systematic offset between VA and PA distributions, which is about 0.16 mm.  
The variation interval for VA lays inside the PA interval. Since the real process variates more than its simulation, an averaging or smoothing effect due to CT measurement and VA algorithm is assumed. As the variation of the PA complies with the process requirements, the VA variation complies as well. In this comparison of distributions only probabilistic statements can be given. Thus, in a spatially resolved comparison, the compliance for each single measurement point is evaluated.