## 4 Deformation and Stress Analysis Using Numerical Modelling  
Experimental measurements showed that stresses on the right and left sides of both painted and unpainted roofs showed significant deviations. The aim of numerical modelling was to show that with symmetrical geometry of the roof structure and the attachment of the antenna on the axis of symmetry, uniform loads are assumed on the right and left side of the roof. For the numerical simulation by the finite element method, a real roof model was used. As already mentioned in Section 2, the control of a real model with a computer model is currently realized in the real part by 3D scanning of shape and dimensions. However, it is necessary to consider what the device is measuring and what are the measured values compared to, or how it should be proceeded in case of optimization of the production process, etc. We do not address this part in this paper. Numerical modelling can, e.g., analyze not only the overall displacements but also their components. Numerical modelling is highly used in the automotive industry. It is a modern and fast method for the implementation of design changes. It is used for different parts of vehicles or different areas of interest--e.g., in [17] the noise and vibration of a roof is investigated using the finite element method and is compared by an experimental modal analysis.  
Due to the fact that it was not necessary to address the safety of the structure itself--the results were to be used to illustrate a possible comparison of the results--the boundary conditions were simplified. Only the part of the roof around the mounting hole attached around the perimeter was analyzed. The static analysis of the roof has been realized by ANSYS software. The created mesh consists of approximately 31,092 volume finite elements with quadratic approximation and approximately 58,405 nodes. A unit load was applied at the antenna site.  
Figure 14: Directions of normal stresses rotated by angle \(\varphi\) (roof C).  
Figure 15 shows a field of resulting displacements, and Figure 16 shows a fields of reduced stresses according to the [PERSON] theory.  
In both cases, the figures show an even distribution of the examined parameters on the right and left side of the roof. This fact corresponds to a theoretical consideration. In our case, however, the displacement field determined by FEM (Figure 15) does not correspond to Figure 5, where the displacements were measured with a Leica high-precision 3D scanning device on a real roof. The values of reduced stresses in places of applied strain gauges determined by numerical modelling do not correspond to values determined for antenna mounting (see Table 3). It should be noted here that due to the defined boundary conditions, the specific values of reduced stresses were not evaluated, but their percentage on the right and left side of the roof and in places 1 and 2 were. In the case of numerical modelling, the differences did not exceed 2%. This deviation is related to the setting of the finite element mesh parameters. For the stress analysis of the antenna assembly process, it is necessary to use a real model of the antenna consisting of several moving parts. From a technical point of view, this is not a problem, but the differences between experimental measurements and numerical modelling may also be related to the accuracy of the shape and dimensions of the antenna used.