Further Development of Vector Generalized Finite Element Method and Its Hybridization With Boundary Integrals

Recently, vector generalized finite element method (VGFEM) was introduced for the solution of the vector Helmholtz equation, and its applicability was validated for canonical problems. VGFEM uses a local Helmholtz decomposition to construct basis functions in overlapping local domains of some canonical shape. While using a canonical shape for local domains adds flexibility to the method, one needs to provide information regarding boundaries of domains/inhomogeneities. The need for surface information proves to be a bottleneck in using the method for a larger class of problems. This paper is targeted towards overcoming these deficiencies; here, we will introduce the modifications to this method that permit interfacing with arbitrarily shaped local domains (to facilitate interfacing with existing meshing software), integrate this method with boundary integrals and provide a framework for studying dispersion. As will be apparent, the hybridization of the method with boundary integrals is not a simple adaptation of existing methods onto the VGFEM framework. Likewise, dispersion analysis is nontrivial due to the overlapping nature of VGFEM basis functions. A range of practical problems has been analyzed within the presented framework and results are compared either against measurements or existing FEM data to validate the presented methodology.