An interval finite element approach for the calculation of envelope frequency response functions

This paper focusses on the application of the interval finite element method in dynamic analyses. It describes a methodology for calculating frequency response function envelopes from a finite element model containing imprecise parameters defined as interval uncertainties. The resulting envelope functions give a conservative approximation of the possible range of the frequency response function, taking into account that the uncertain parameters in the model can adopt any value in their presumed uncertainty intervals. The methodology is based on the modal superposition principle. It consists of an interval aritmethic algorithm which processes the results of a preliminary global optimization performed on the modal parameters. The algorithm is constructed such that it optimally combines the advantages of both the anti-optimization and the interval arithmetic strategy for general numerical interval calculations. In the first stage of the development, the modal parameter ranges of each individual mode are independently combined in the modal response contributions. This yields the modal rectangle (MR) method. In order to remedy the high conservatism inherent to the MR method, the exact eigenfrequency ranges are added to the analysis. This results in the modal rectangle method with eigenfrequency interval correction (MRE). A second improvement consists of adding extra delimiters to the MRE modal parameter range approximation. This is achieved by performing an extra optimization on the modal response contributions at discrete frequencies. The method is referred to as the locally optimized modal rectangle method with eigenfrequency interval correction (OMRE). Finally, a numerical example illustrates the different algorithms