3D Extended Isogeometric Boundary Element Method (XIBEM) for Acoustic Wave Scattering.

An isogeometric boundary element method based on NURBS is used to find solutions to the Helmholtz equation. The method is extended in a partition-of-unity sense, multiplying the NURBS functions with families of plane waves; this new method is called the eXtended Isogeometric Boundary Element Method (XIBEM). When compared to non-enriched boundary element simulations, using XIBEM significantly reduces the number of degrees of freedom required to obtain a solution with a given error. The extension used here (and in similar Tretz methods) requires a set of wave directions to be specified. Ideally, these directions are uniformly spaced points over the unit-sphere. Simple schemes (e.g. latitude/longitude and discretised cube methods) have been proposed previously. However, while these schemes provide adequate spacing of wave directions, they do not allow an arbitrary number of wave directions to the be chosen. The authors use a novel algorithm, based on a physical analogy of charged particles held in static equilibrium on a spherical surface. The XIBEM formulation will be described, including a focus on the novel method of choosing a uniformly spaced set of plane wave directions for the enrichment. Numerical results show the reduction in degrees of freedom required to obtain approximations of engineering accuracy