Hybrid method for modelling anti-plane vibrations of layered waveguides with bonded composite joints

Abstract

The use of mesh-based methods for modeling elongated composite structures with inhomogeneities leads to an increase in computational costs when discretizing the waveguide part of greater linear dimensions, while the semi-analytical numerical methods do not allow one to directly describe the structures with local inhomogeneities of an arbitrary shape. To compensate for the shortcomings of these two classes of numerical methods, we propose a hybrid scheme based on the spectral element method (SEM) and the semi-analytical finite element method (SAFEM) for studying the anti-plane vibrations of a composite structure in the frequency domain. Thus, for a waveguide, this scheme makes it possible to represent the solution via the SAFEM as a sum of modes or guided waves, and the adjacent regions are discretized using the SEM. The displacement and stress continuity conditions are imposed on the common boundary of two domains. To couple the solutions, we introduce an auxiliary function of displacement, which is approximated by applying the same basis functions as those used in SEM and SAFEM (Lagrange interpolation polynomials on the Gauss–Legendre–Lobatto nodal points). The unknown expansion coefficients of this function are determined by the Galerkin and collocation methods. It has been established that both methods provide the same accuracy. The results obtained by the hybrid approach employing the Galerkin and collocation methods are compared with the results calculated in the standard finite element software. It is shown that they are in good agreement. The presented hybrid approach can be straightforwardly extended to the case of in-plane motion, but it requires significant refinement for a three-dimensional case.