# Stiffness Estimation and Equivalence of Boundary Conditions in FEM Models
## 5 Conclusions  
In the paper, the methods of the equivalence of boundary conditions in the FE model were described. Their basis is the replacement of boundary condition, bond, constraint, or a part of the model by bushing or elastic connectors with corresponding stiffness. The paper presents two approaches to determine the stiffness parameters. The first method is based on a purely numerical calculation and is applicable when a real physical model does not exist. This approach is very accurate and effective. The second approach is based on the finite element model updating, where the stiffness parameters are determined using parametric optimization and modal analysis. The accuracy of this technique depends on several factors, such as natural frequencies, material properties, dimensions, and geometry of the structure, on the basis of which the FE model is created. From this point of view, it is an estimation method. However, the test results indicate that the estimate is relatively accurate and the procedure is practically applicable. It has also been shown that, in the case of replacing part of the model, the problem is significantly simplified, which is positively reflected in the shortening of the computational time.  
While the first method is relatively simple and can be easily applied, the second method requires knowing the natural frequencies of the structure. These can be determined by experimental modal analysis. There are several measurement techniques that can be used for this purpose. In addition to conventional methods with accelerometers, advanced contactless optical methods can be used to perform either spot measurements or full-field measurements. Let us mention, for example, scanning laser Doppler vibrometry or high-speed digital image correlation. The advantage of experimental modal analysis is that it also provides information on structural damping. This can be used for advanced updating of FE models in dynamic analyses. In case more complex structures are analyzed, it is necessary to perform a multi-reference modal analysis to capture both global and local modes. If the structure is too complex to be measured effectively, the natural frequencies can be determined by numerical modal analysis. The weakness of the method is parametric  
Figure 12: Bumper FE model.  
optimization with one objective function. On the other hand, the accuracy of the method proved to be sufficient. In addition, the method has been shown to be applicable even if not all natural frequencies are included in the optimization process. From this point of view, this procedure for determining equivalent stiffness parameters can be helpful for CAE engineers and researchers who do not have advanced software for multi-objective optimization.  
Conceptualization, R.H.; data curation, P.L.; formal analysis, P.P. and J.M.; funding acquisition, R.H.; investigation, P.P. and A.K.; methodology, R.H. and P.L.; project administration, P.L.; resources, R.H. and P.L.; software, P.L. and A.K.; supervision, R.H.; validation, R.H.; visualization, A.K. and J.M.; writing--original draft, R.H. and P.P.; writing--review and editing, R.H. and P.L. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Scientific Grant Agency of the Slovak Republic under grant VEGA 1/0355/18 and by Research and Development Operational Program funded by the ERDF within the project University Science Park TECHNICOM for Innovation Application Supported by Knowledge Technology, ITMS: 2622020182.  
Not applicable.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.