Weak Form Implementation of the Semi-Analytical Finite Element (SAFE) Method for a Variety of Elastodynamic Waveguides

Dispersion curves are essential to any guided wave NDE project. The Semi-Analytical Finite Element (SAFE) method has significantly increased the ease by which these curves can be calculated. However, due to misconceptions regarding theory and fragmentation based on different finite-element software, the theory has stagnated, and adoption by researchers who are new to the field has been slow. This paper focuses on the relationship between the SAFE formulation and finite element theory, and the implementation of the SAFE method in a weak form for plates, pipes, layered waveguides/composites, curved waveguides, and arbitrary cross-sections is shown. The benefits of the weak form are briefly described, as is implementation in open-source and commercial finite element software. The purpose of this paper is to inform researchers of the state-of-the-art from a slightly different perspective and demonstrate an alternative to writing their own SAFE code from scratch. Examples given are done using COMSOL Multiphysics Finite Element software, but open-source options are described as well. The scope of this paper is limited to waveguides with traction-free boundaries.