In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial boundaries for truncated domains. We consider three different variational forms for this problem, depending on the functional space for the solution, in particular, in what refers to the regularity required on artificial boundaries. Then, stabilized finite element methods that can mimic these three functional settings are described. Stability and convergence analyses of these stabilized formulations including the NRBC are presented. Additionally, numerical convergence test are evaluated for various polynomial interpolations, stabilization methods and variational forms. Finally, several benchmark problems are solved to determine the accuracy of these methods in 2D and 3D.
Abstract
In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial boundaries [...]
In this work we solve the compressible Navier–Stokes equations written in primitive variables in order to simulate low Mach number aeroacoustic flows. We develop a Variational Multi-Scale formulation to stabilize the finite element discretization by including the orthogonal, dynamic and non-linear subscales, together with an implicit scheme for advancing in time. Three additional features define the proposed numerical scheme: the splitting of the pressure and temperature variables into a relative and a reference part, the definition of the matrix of stabilization parameters in terms of a modified velocity that accounts for the local compressibility, and the approximation of the dynamic stabilization matrix for the time dependent subscales. We also include a weak imposition of implicit non-reflecting boundary conditions in order to overcome the challenges that arise in the aeroacoustic simulations at low compressibility regimes. The order of accuracy of the method is verified for two- and three-dimensional linear and quadratic elements using steady manufactured solutions. Several benchmark flow problems are studied, including transient examples and aeroacoustic applications.
Abstract
In this work we solve the compressible Navier–Stokes equations written in primitive variables in order to simulate low Mach number aeroacoustic [...]