Helmholtz equation and non-singular boundary elements applied to multi-disciplinary physical problems

The famous scientist Hermann von Helmholtz was born 200 years ago. Many complex physical wave phenomena in engineering can effectively be described using one or a set of equations named after him: the Helmholtz equation. Although this has been known for a long time from a theoretical point of view, the actual numerical implementation has often been hindered by divergence free and/or curl free constraints. There is further a need for a numerical method which is accurate, reliable and takes into account radiation conditions at infinity. The classical boundary element method (BEM) satisfies the last condition, yet one has to deal with singularities in the implementation. Since these singularities are mathematical in origin, they can actually be removed without losing accuracy by subtracting a carefully chosen theoretical solution with the same singular behavior. We review here how a recently developed singularity-free 3D boundary element framework with superior accuracy can be used to tackle such problems only using one or more Helmholtz equations with higher order (quadratic) elements which can tackle complex shapes. Examples are given for acoustics (a Helmholtz resonator among others) and electromagnetic scattering. We briefly touch on the Helmholtz decomposition for dynamic elastic waves as well.