Wave boundary elements: a theoretical overview presenting applications in scattering of short waves

Abstract

It is well known that the use of conventional discrete numerical methods of analysis (FEM and BEM) in the solution of Helmholtz and elastodynamic wave problems is limited by an upper bound on frequency. The current work addresses this problem by incorporating the underlying wave behaviour of the solution into the formulation of a boundary element, using ideas arising from the Partition of Unity finite element methods. The resulting ‘wave boundary elements’ have been found to provide highly accurate solutions (10 digit accuracy in comparison with analytical solutions is not uncommon). Moreover, excellent results are presented for models in which each element may span many full wavelengths. It has been found that the wave boundary elements have a requirement to use only around 2.5 degrees of freedom per wavelength, instead of the 8–10 degrees of freedom per wavelength required by conventional direct collocation elements, extending the supported frequency range for any given computational resources by a factor of three for 2D problems, or by a factor of 10–15 for 3D problems. This is expected to have a significant impact on the range of simulations available to engineers working in acoustic simulation. This paper presents an outline of the formulation, a description of the most important considerations for numerical implementation, and a range of application examples.