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 that 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. We review here how a recently developed singularity-free three-dimensional (3D) boundary element framework with superior accuracy can be used to tackle such problems only using one or a few Helmholtz equations with higher order (quadratic) elements which can tackle complex curved shapes. Examples are given for acoustics (a Helmholtz resonator among others) and electromagnetic scattering.
翻译:著名的科学家Hermann von Helmholtz是200年前诞生的。工程中许多复杂的物理波现象可以用以他命名的一个或一组方程式来有效地描述:Helmholtz方程式。虽然从理论观点看,人们早就知道这一点,但实际的数值执行往往受到自由差异和/或无曲线限制的阻碍。还需要一种精确、可靠和考虑到无限辐射条件的数字方法。典型的边界要素方法(BEM)满足了最后的条件,但必须处理执行过程中的奇特性。我们在这里审查最近开发的无奇特性三维(3D)边界要素框架如何只能用一个或几个具有更高顺序(二次)的赫尔姆霍茨方程式来处理这类问题,这些方程式可以处理复杂的曲线形状。例如,声学(一个HelmholtzResonator等)和电磁散射。