Hybrid FE-SAFE modelling of elastic wave scattering in curved waveguides

A 3D hybrid finite element/semi-analytical finite element (FE-SAFE) method is developped to investigate elastic wave scattering in curved waveguides. This method is inspired from hybrid models already proposed in the literature for 2D straight waveguides see for instance Karunasena et al., Comput. Meth. Appl. Mech. Eng. 125, 221-233 (1995). It consists in analysing near field inhomogeneities (cracks for instance) with the standard FE method, while the solution is expanded as sums of guided modes on the inlet cross-section (and outlet, if any). Such an eigenmode expansion allows to separate ingoing from outgoing waves. In the hybrid FE-SAFE model, the amplitudes of the ingoing modes (incident) are enforced. Those of the outgoing scattered modes (reflected, transmitted) are solution of the hybrid system. Concerning the computation of guided modes in curved waveguides, a SAFE technique that has been previously developed by the author is used. The SAFE variational formulation must be rewritten in a helical coordinate system, translationally invariant along the waveguide axis. From eigendisplacements and SAFE matrices, direct formula are proposed for the computation of the energy velocity, power flow and load eigenvectors. This avoids the tedious post-processing step of calculating consistent load eigenvectors with hybrid FE-SAFE methods. The proposed formula are likely to simplify the development of hybrid methods inside existing FE softwares. Wave scattering inside a free-end 'twisted cylinder' is first considered for validation purpose. A comparison with literature results obtained for the free-end cylinder is performed. Some results are then presented for fundamental modes of helical waveguides interacting with a free-end boundary (complete crack). The power balance is satisfied with good accuracy. The hybrid method is implemented in a generic in-house 3D FE code developed under Matlab. The main advantage of this fully numerical approach is that complex-shaped inhomogeneities and waveguides of arbitrary cross-section can be handled