Abstract

In this paper a computational methodology leading to the development of a new class of FEs, based on the application of continuous and smooth approximation polynomials, being splines, has been presented. Application of the splines as appropriately defined piecewise elemental shape functions led the authors to the formulation of a new approach for FEM, named as spFEM, where contrary to the well-known NURBS approach, the boundaries of spFEs are well-defined, exactly as it is in the case of the traditional FEM. The current approach has been computationally verified by the authors it terms of high frequency dynamics including such problems as: spectra of natural frequencies, modes of natural vibrations as well as wave propagation problems, especially in the aspect of high frequency responses, all in the case of selected problems involving one- and two-mode theories of 1-D structural elements. The applicability of the proposed approach has been evaluated and compared, in terms of calculated dynamic responses, with the results obtained by the use of well-established FEM approaches: classical FEM as well as TD-SFEM. In all cases investigated by the authors the proposed spFEM approach turned out to be the most accurate approach, free from the main drawback of the other tested FEM approaches thanks to the class of differentiability of approximation polynomials, which guarantees the absence of frequency band gaps in calculated spectra of natural frequencies. A direct consequence of this feature of the proposed approach is that a larger part of the calculated spectra of natural frequency, the same as modes of natural vibrations, can effectively be used for more accurate calculations of dynamic reposes even in the case of multi-mode theories. This in contrast to the other tested FEM approaches.

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