Higher‐order surface treatment for discontinuous Galerkin methods with applications to aerodynamics

Abstract

SUMMARYWhen dealing with high-order numerical methods, an adequate treatment of curved surfaces is required not only to guarantee that the expected high-order is maintained in the vicinity of surfaces but also to avoid steady-state convergence issues. Among the variety of high-order surface treatment techniques that have been proposed, the ones employing NURBS (non-uniform rational B-splines) to describe curved surfaces can be considered superior both in terms of accuracy and compatibility with computer-aided design softwares. The current study describes in detail the integration of NURBS-based geometry description in a high-order solver based on the discontinuous Galerkin formulation. Particularly, this work also discusses how and why NURBS curves of very high order can be employed within standard NURBS-based boundary treatment techniques to yield reduced implementation complexity and computational overhead. Theoretical estimates are provided along with numerical experiments in order to support the proposed approach. Minding engineering applications in the context of compressible aerodynamics, additional simulations are addressed as numerical examples to illustrate the advantages of using higher-order NURBS in practical situations.

CAD‐consistent adaptive refinement using a NURBS‐based discontinuous Galerkin method

“…In particular, the reflected shocks are well captured, as well as the interactions with the vortex shedding process. As can be seen in table (2), this improvement is obtained for a significant increase of the number of elements, which remains however far lower than that of the uniform fine grid. Nevertheless, some discrepencies with the reference results can still be observed in the region where shocks and vortices interact.…”

“…Therefore, the smallest elements have the same size in the two cases. The grid parameters are summarized in table (2). The parameter controling the artificial viscosity switch is set to the values 0 = 5.…”

“…Consequently, points at the boundary of the grid do not lie exactly on the CAD geometry, normals and derivatives are not preserved. More advanced approaches have been proposed recently, to fully integrate NURBS geometries in DG solvers, independently from the approximation space [2,28]. The demonstrated results are convincing, in terms of accuracy and convergence, but the proposed methodology may suffer from the complexity of the underlying mappings used for spatial integration.…”

“…Besides, several recent studies have pointed out the limitations related to the use of piecewise linear grids and the necessity to use curved meshes to obtain the theoretical convergence rates. The use of a piecewise linear approximation of the geometry may also lead to spurious phenomena in the solution [1,2,3]. This is especially true in the context of high-order schemes based on Discontinuous Galerkin (DG) or Finite-Volume (FV) methods, which are more and more employed to solve complex problems, for instance in Large Eddy Simulation (LES).…”

Foundations of the blended isogeometric discontinuous Galerkin (BIDG) method

“…The BIDG method seamlessly combines exact geometric design (CAD) with high-order accurate analysis (DGFEM), as schematically shown in figure 1. Some groundwork for BIDG-type methods is provided in [3,6,47,55,57], with patchbased methods recently being presented in [40,41,72] for elliptic problems. Also, please note the remark at the end of the appendix for a brief discussion of what is exactly meant by "isogeometric" throughout this paper.…”