362 S. Tornincasa et al. Concerning Finite Element Analysis, widely used to perform the parametric analyses, a new technique named Isogeometric Analysis (IGA) [9] is being developed with the aim of making design (mostly Computer Aided Design) and simulation (FEM) converge to the same framework and take advantage of the integration of the two environments to manage the change in the geometrical definition to the mathematical model, by using the Non-Uniform Rational B-Splines (NURBS) function that define the geometry, directly as basis function in the FE model. Since its first results, IGA has shown giving several advantages in structural mechanics such as smoothness across the element boundaries and the possibility of raising the order and gaining smoothness [10], better performance in the simulation of problems with contacts [11–15] having smooth interface and not gaps and overlaps, due to its tight relationship with CAD it allows to perform topological optimization directly on the geometrical model [16–19]. In the field of structural vibrations, IGA provides advantageous properties, mostly due to the gain of accuracy at high orders, with opposite behaviour compared to the high-order standard elements, and the positiveness of the entries of the mass matrix, due to the non-negative property of NURBS basis functions, which give more stability in transient dynamics problems. NURBS-based IGA has some disadvantages which are topics for recent and future developments of the methods, such as the local or adaptive refinement and the need of using several NURBS patches for a certain complexity of the geometry. T-Spline [20, 21] is one of the technologies that could replace NURBS and solve both the problems, but a trivariate version is not yet established in CAD software packages. For this reason a multi-patch geometry is considered for modelling components with complex shape, but it raise the problem of connecting the patches together. In case of the possibility to generate a conforming parameterization at the patch interface, a simple node-to-node master–slave relationship can be defined and the implementation is straightforward, but when this is not possible the coupling can be performing using Nitsche’s method [22]. This method is known for the imposition of boundary conditions in the weak form, and an extension can perform the coupling of different domains. The method is in between the Lagrange multipliers and the penalty method, and it takes the advantage of both, namely the consistency of the Lagrange multiplier approach, and the relative ease of implementation and and parameter selection of the penalty method. The surveys by Imregun and Visser [23], in the 1990s, show the emerging of finite element model updating. The problem of updating a numerical model by using data acquired from a physical vibration test is richly handled by Mottershead and Friswell [24]. They showed how many issues are to be addressed to produce the desired improvement. The methods are either direct or iterative. The latter are based on minimizing an objective function that is generally a non-linear function. The effect of the improvement due to including second order sensitivities was studied by Kim et al. [25]. Another type of method was proposed by Lin et al. [26] to employ both the analytical and the experimental modal data for evaluating sensitivity coefficients with the objective of improving convergence to cases where there is a higher error magnitude, which happens for complex analytical model. Although several different approaches have been proposed and successfully applied on different structures, the authors of this paper would propose their experience on avionic equipment. The aim is to define an appropriate and updated FE model for modal analysis (FEA) and to match the numerical results with an experimental modal analysis (EMA) campaign. The inverse eigensensitivity approach [24, 27] is proposed as an iterative model updating technique with respect to an experimental modal test campaign. Its validity and quick convergence has been demonstrated in the literature, but modal truncation and experimental inaccuracy effects may represent an interesting task for model updating of this kind of structures. The principal task of this article is the comparison of two different version of the method used to reach a good matching between the modal characteristics of an avionic structure modeled with a linear FE approach and substructured in components. The mathematical bases are the same for both methods but the difference is defined in the objective functions to minimize, and this difference goes to affect the dimensions of the sensitivity matrix that are smaller for the new method than for the previous one. In this paper an experimental test-rig already considered in [28] is used as subject to investigate its crossing and veering phenomena, focusing on the modelling of the test-rig using NURBS, in order to obtain an analysis-suitable geometry, with the properties suitable for running a structural modal isogeometric analysis. In the first section, the test-rig and its aims and features are presented, in order to underline some of the characteristics to focus the attention to. In the second section, NURBS and isogeometric analysis in general are considered, with description of the main concept and the differences with respect to standard FEA, and the NURBS model of the test-rig is presented as well, with all the details to understand how it is modelled. In the third section it is presented the concept of Nitsche’s method for domain coupling, with details of the entries of the coupling terms and matrices, with the application to the test-rig.
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