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

Summary When 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. Copyright © 2015 John Wiley & Sons, Ltd.

[1]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[2]  Ruben Sevilla,et al.  Numerical integration over 2D NURBS-shaped domains with applications to NURBS-enhanced FEM , 2011 .

[3]  F. Brezzi,et al.  Discontinuous Galerkin approximations for elliptic problems , 2000 .

[4]  George Em Karniadakis,et al.  The Development of Discontinuous Galerkin Methods , 2000 .

[5]  Per-Olof Persson,et al.  A Simple Mesh Generator in MATLAB , 2004, SIAM Rev..

[6]  J. Peraire,et al.  Sub-Cell Shock Capturing for Discontinuous Galerkin Methods , 2006 .

[7]  O. Zienkiewicz.  The Finite Element Method In Engineering Science , 1971 .

[8]  David L. Darmofal,et al.  Shock Capturing with Higher-Order, PDE-Based Artificial Viscosity , 2007 .

[9]  Antonio Huerta,et al.  Comparison of high‐order curved finite elements , 2011 .

[10]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[11]  Antony Jameson,et al.  Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists , 2011 .

[12]  Z. Wang.  High-order methods for the Euler and Navier–Stokes equations on unstructured grids , 2007 .

[13]  G. Karniadakis,et al.  Spectral/hp Element Methods for Computational Fluid Dynamics , 2005 .

[14]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[15]  S. Rebay,et al.  High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations , 1997 .

[16]  Les A. Piegl,et al.  The NURBS book (2nd ed.) , 1997 .

[17]  Rainald Löhner,et al.  A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids , 2007, J. Comput. Phys..

[18]  John A. Ekaterinaris,et al.  High-order accurate, low numerical diffusion methods for aerodynamics , 2005 .

[19]  William J. Rider,et al.  The use of classical Lax–Friedrichs Riemann solvers with discontinuous Galerkin methods , 2001 .

[20]  R. Cárdenas.  NURBS-Enhanced Finite Element Method (NEFEM) , 2009 .

[21]  Antonio Huerta,et al.  NURBS‐enhanced finite element method for Euler equations , 2008 .

[22]  George Em Karniadakis,et al.  Spectral/hp Methods for Viscous Compressible Flows on Unstructured 2D Meshes , 1998 .

[23]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[24]  L. Trefethen.  Six myths of polynomial interpolation and quadrature , 2011 .

[25]  Lilia Krivodonova,et al.  High-order accurate implementation of solid wall boundary conditions in curved geometries , 2006 .

[26]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[27]  Ralf Hartmann,et al.  Discontinuous Galerkin methods for computational aerodynamics — 3D adaptive flow simulation with the DLR PADGE code , 2010 .

[28]  Les A. Piegl,et al.  On NURBS: A Survey , 2004 .

[29]  R. Moura,et al.  LYAPUNOV EXPONENTS AND ADAPTIVE MESH REFINEMENT FOR HIGH-SPEED FLOWS , 2013 .