On wave based Trefftz and weak Trefftz discontinuous Galerkin approaches for medium-frequency heterogeneous Helmholtz problem

Medium frequency heterogeneous Helmholtz problem gives rise to severe challenge to existing numerical approaches. Two methods are developed to solve this problem, namely extended Variational Theory of Complex Rays(VTCR) method and wave based weak Trefftz discontinuous Galerkin(WTDG) method. The VTCR belongs to Trefftz method, where general solutions of governing equation are used as shape functions to get approximate solutions for problem in vibroacoustics. When the square of wave number varies linearly in the media, one considers the extended VTCR with shape functions namely Airy wave functions. Then a general way to handle heterogeneous media by the wave based WTDG is proposed. There is no a priori restriction for the wave number. One locally develops general approximate solution of the governing equation, the gradient of the wave number being the small parameter. In this way, zero order and first order approximations are defined, namely Zero Order WTDG and First Order WTDG. Their shape functions only satisfy the local governing equation in average sense. It proves that the extended VTCR and the wave based WTDG propose an efficient and flexible way to solve problem in vibroacoustics.