Overset meshing coupled with hybridizable discontinuous Galerkin finite elements

Abstract

Summary We introduce the use of hybridizable discontinuous Galerkin (HDG) finite element methods on overlapping (overset) meshes. Overset mesh methods are advantageous for solving problems on complex geometrical domains. We combine geometric flexibility of overset methods with the advantages of HDG methods: arbitrarily high‐order accuracy, reduced size of the global discrete problem, and the ability to solve elliptic, parabolic, and/or hyperbolic problems with a unified form of discretization. Our approach to developing the ‘overset HDG’ method is to couple the global solution from one mesh to the local solution on the overset mesh. We present numerical examples for steady convection–diffusion and static elasticity problems. The examples demonstrate optimal order convergence in all primal fields for an arbitrary amount of overlap of the underlying meshes. Copyright © 2017 John Wiley & Sons, Ltd.

An Overset Mesh Framework for an Isentropic ALE Navier-Stokes HDG Formulation

“…. , N. We have implemented the overset coupling algorithm that is described in more detail in Kauffman et al [2], and have utilized a sparse-direct linear solver UMFPACK [30] to obtain the global solution through the deal.II finite element library [31].…”

“…This scheme decomposes the computational domain into a set of structured overset meshes, which must at least abut to avoid gaps and voids in the domain. Communication via interpolation only occurs on the overset boundary faces, which is the same approach we are employing here and we first presented in [2]. The DG Chimera scheme also can handle mixed order meshes, so interpolation occurs between meshes of different approximation orders.…”

“…In this paper we are extending the overset mesh framework originally developed in [2] to an isentropic ALE Navier-Stokes HDG formulation. This formulation allows for bodies to move throughout the domain, relative to one another, and for the computational domain to deform.…”

“…This last point is the driving motivation for utilizing an overset mesh framework. We have previously developed an overset mesh framework with the mathematical discretization of the hybridizable discontinuous Galerkin (HDG) finite element method for static linear problems such as convection-diffusion and elastostatics [2]. Below we present motivation for the use of an overset mesh method and the HDG method.…”

A robust overset assembly method for multiple overlapping bodies

“…In terms of the integration overset and solver, the University of Maryland significantly improved the convergence by using O‐GMRES in the simulation's inner step, 15 and the GARFIELD solver simulation that uses a mixed CPU/GPU approach was also developed 16 . Kauffman and Brazell conducted research on the discontinuous Galerkin (DG) method, 13,17 Jacobs studied the higher order scheme, 18 and French developed the Dirichlet boundary for oversets 19 . Chan conducted overset grid quality diagnostics, 20 and Quon used radial basis functions (RBFs) to avoid the influence of orphan points on the simulation 21 .…”