# Stiffness Estimation and Equivalence of Boundary Conditions in FEM Models
## 1 Introduction  
Numerical methods are a highly effective tool for solving complex problems in the field of structural mechanics, acoustics, fluid flow, heat transfer, mass transfer, electromagnetism, multiphysics, etc. They have become an integral part of the daily work of engineers, scientists and researchers. The finite element method (FEM) is the most widely used method of them. The correctness of the numerical solution in the finite element method is largely dependent on the accuracy of the FE model itself. The basis is to correctly define material properties, initial conditions, loads, and boundary conditions such as constraints, connections, and contacts in accordance with the real ones. Any inaccuracies in the model definition lead to deviations or errors in the load response. In particular, dynamic problems, such as modal analysis [1, 2], impact tests [3, 4], and fatigue analyses [5, 6], are quite sensitive to boundary conditions and material parameters. The most common causes of differences in model behavior may be constraint stiffness and structural damping [7, 8]. Computational models are usually verified by correlating the results of numerical calculation and experimental measurement. A suitable approach is to compare modal parameters (i.e., natural frequencies and mode shapes), for example, using the Modal Assurance Criterion (MAC) [9, 10, 11]. In order to improve the FE model, it is then possible to perform a sensitivity analysis or an optimization analysis, where the response is adjusted by changing the input parameters or identifying them. This procedure is known as finite element model updating. The finite element model updating procedure was introduced in the 1980s [12]. Its main purpose is to calibrate the FE model of an analyzed structure in an appropriate way in order to match numerical and experimental results. Based on the correlation of theseresults, it is possible to identify inaccuracies or determine unknown parameters in the initial finite element model, such as the material properties, boundary conditions, etc. From this point of view, the FE model updating is an inverse task. A number of FE model updating methods have been proposed. They can be divided into non-iterative (direct) methods and iterative methods. The direct methods update the elements of stiffness and mass matrices using the equations of motion and the orthogonality equations [13]. The iterative methods typically involve using the sensitivity of the parameters to find their changes [14]. They are usually posed as an optimization problem, in which an objective function represents the model error (i.e., the difference between the numerical and experimental response). Most of FE model updating methods are based on frequency response functions (FRFs) or modal parameters, which are usually determined by experimental or operational modal analysis. Therefore, they are referred to as vibration-based model updating methods. The number of publications devoted to FE model updating proves that it is a robust tool widely used in many areas, including the design, mechanical engineering, civil engineering, automotive, aerospace, and so on. In addition to refining FE models, it can also be used for system parameter identification, damage detection, health monitoring, structural control, and to determine unknown or equivalent material properties. [PERSON] et al. [15] used the model updating method based on multi-objective optimization to identify joints parameters by approaching actual test results. Their methodology was successfully tested on the SUV frame with seam and spot welds. [PERSON] and [PERSON] [16] proposed the method that is able to avoid the need to achieve an idealized boundary condition when performing experimental tests for model updating purposes. That is, the FE model updating can be achieved in the presence of an arbitrary or unknown boundary condition using in-situ measurements.