Full Article - Open Access.

Idioma principal


Ceze, Marco; Fidkowski, Krzysztof;

Full Article:

We present a method for concurrent mesh and polynomial-order adaptation with the objective of direct minimization of output error using a selection process for choosing the optimal refinement option from a discrete set of choices that includes directional spatial resolution and approximation order increment. The scheme is geared towards compressible viscous aerodynamic flows, in which solution features make certain refinement options more efficient compared to others. No attempt is made, however, to measure the solution anisotropy or smoothness directly or to incorporate it into the scheme. Rather, mesh anisotropy and approximation order distribution arise naturally from the optimization of a merit function that incorporates both an output sensitivity and a measure of the computational cost of solving on the new mesh. An adjoint state is used to translate the residual perturbation resulting from each refinement option into an output sensitivity with respect to each mesh modification option. Two measures of computational cost are explored: a generic measure that accounts for the number of degrees of freedom of the discrete state, and one that accounts for the number of floating-point operations involved in solving the discrete problem. We restrict the mesh refinement mechanics to quadrilateral and hexahedral meshes. Many such meshes and associated meshing programs exist from the structured CFD community, and these can be leveraged to produce the starting meshes for the proposed adaptation. Additionally, we discuss implementation challenges of hp-adaptive methods for aerodynamic problems, such as load balancing on distributed-memory systems. The method is applied to output-based adaptive simulations of laminar and Reynolds-averaged compressible Navier-Stokes equations on body-fitted meshes in two and three dimensions. Two-dimensional results show significant reduction in the degrees of freedom and computational time to achieve output convergence when the discrete choice optimization is used compared to uniform h or p adaptation. Threedimensional results show that the presented method is an affordable way of achieving output convergence on notoriously difficult cases such as the third Drag Prediction Workshop W1 configuration.

Full Article:

Palavras-chave: hp-adaptation, computational fluid dynamics, discontinuous Galerkin.,


DOI: 10.5151/meceng-wccm2012-18038

Referências bibliográficas
  • [1] Timothy Barth and Mats Larson, A posteriori error estimates for higher order Godunov finite volume methods on unstructured meshes, Finite Volumes for Complex Applications III (London) (R. Herban and D. Kröner, eds.), Hermes Penton, 2002, pp. 41–63.
  • [2] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numerica (A. Iserles, ed.), Cambridge University Press, 2001, pp. 1–10
  • [3] Kim S. Bey, An hp-adaptive discontinuous Galerkin method for hyperbolic conservation laws, Ph.D. thesis, University of Texas at Austin, 1994.
  • [4] Nicholas K. Burgess and Dimitri J. Mavriplis, An hp-adaptive discontinuous Galerkin solver for aerodynamic flows on mixed-element meshes, 49th AIAA Aerospace Sciences Meeting and Exhibit, 2011.
  • [5] M. J. Castro-Diaz, F. Hecht, B.Mohammadi, and O. Pironneau, Anisotropic unstructured mesh adaptation for flow simulations, International Journal for Numerical Methods in Fluids 25 (1997), 475–491.
  • [6] Marco Ceze and Krzysztof J. Fidkowski, Output-driven anisotropic mesh adaptation for viscous flows using discrete choice optimization, 48th AIAA Aerospace Sciences Meeting and Exhibit, 2010.
  • [7] , A robust adaptive solution strategy for high-order implicit CFD solvers, 20th AIAA Computaional Fluid Dynamics Conference, AIAA, 2011.
  • [8] , An anisotropic hp-adaptation framework for functional prediction, AIAA Journal (2012 Accepted).
  • [9] K. J. Fidkowski and D. L. Darmofal, A triangular cut–cell adaptive method for high– order discretizations of the compressible Navier–Stokes equations, Journal of Computational Physics 225 (2007), 1653–1672.
  • [10] Krzysztof J. Fidkowski, A high–order discontinuous Galerkin multigrid solver for aerodynamic applications, MS thesis, M.I.T., Department of Aeronautics and Astronautics, June 2004.
  • [11] , A simplex cut-cell adaptive method for high–order discretizations of the compressible Navier-Stokes equations, PhD dissertation,Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, June 2007.
  • [12] Krzysztof J. Fidkowski and David L. Darmofal, Review of output-based error estimation and mesh adaptation in computational fluid dynamics, AIAA Journal 49 (2011), no. 4, 673–694.
  • [13] L. Formaggia, S. Micheletti, and S. Perotto, Anisotropic mesh adaptation with applications to CFD problems, Fifth World Congress on Computational Mechanics (Vienna, Austria) (H. A. Mang, F. G. Rammerstorfer, and J. Eberhardsteiner, eds.), July 7-12 2002.
  • [14] Luca Formaggia and Simona Perotto, New anisotropic a priori error estimates, Numerische Mathematik 89 (2001), 641–667.
  • [15] Luca Formaggia, Simona Perotto, and Paolo Zunino, An anisotropic a posteriori error estimate for a convection-diffusion problem, Computing and Visualization in Science 4 (2001), 99–104.
  • [16] Stefano Giani and Paul Houston, High–order hp–adaptive discontinuous Galerkin finite element methods for compressible fluid flows, ADIGMA - A European Initiative on the Development of Adaptive Higher-Order Variational Methods for Aerospace Applications (Norbert Kroll, Heribert Bieler, Herman Deconinck, Vincent Couaillier, Harmen van der Ven, and Kaare Sørensen, eds.), Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 113, Springer Berlin / Heidelberg, 2010, pp. 399–411.
  • [17] M. B. Giles and E. Süli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta Numerica, vol. 11, 2002, pp. 145–236.
  • [18] Michael B. Giles, Mihai C. Duta, Jens-Dominik Müller, and Niles A. Pierce, Algorithm developments for discrete adjoint methods, AIAA Journal 41 (2003), no. 2, 198–205.
  • [19] Wagdi G. Habashi, Julien Dompierre, Yves Bourgault, Djaffar Ait-Ali-Yahia, Michel Fortin, and Marie-Gabrielle Vallet, Anisotropic mesh adaptation: towards userindependent, mesh-independent and solver-independent CFD. Part I: general principles, International Journal for Numerical Methods in Fluids 32 (2000), 725–744.
  • [20] Ralf Hartmann, Adjoint consistency analysis of discontinuous Galerkin discretizations, SIAM Journal on Numerical Analysis 45 (2007), no. 6, 2671–2696.
  • [21] Ralf Hartmann and Paul Houston, Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations, Journal of Computational Physics 183 (2002), no. 2, 508–532.
  • [22] V. Heuveline and R. Rannacher, Duality-based adaptivity in the hp-finite element method, Journal of Numerical Mathematics 11 (2003), no. 2, 95–113.
  • [23] Paul Houston, Emmanuil H. Georgoulis, and Edward Hall, Adaptivity and a posteriori error estimation for DG methods on anisotropic meshes, International Conference on Boundary and Interior Layers, 2006.
  • [24] Paul Houston and Endre Süli, A note on the design of hp-adaptive finite element methods for elliptic partial differential equations, Computer Methods in Applied Mechanics and Engineering 194 (2005), 229–243.
  • [25] G. Karypis and V. Kumar, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM Journal for Scientific Computing 20 (1998), no. 1, 359–392.
  • [26] James Lu, An a posteriori error control framework for adaptive precision optimization using discontinuous Galerkin finite element method, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 2005.
  • [27] Charles A. Mader, Joaquim R.R.A. Martins, Juan J. Alonso, and Edwin van der Weide, ADjoint: An approach for the rapid development of discrete adjoint solvers, AIAA Journal 46 (2008), no. 4, 863–873.
  • [28] D. J. Mavriplis, Adaptive mesh generation for viscous flows using Delaunay triangulation, Journal of Computational Physics 90 (1990), 271–291.
  • [29] D. J. Mavriplis and A. Jameson, Hermite-based mesh adaptation for functional outputs improvement in fluid flow simulation, AIAA Journal 47 (2009), no. 8, 1965–1976.
  • [30] Todd A. Oliver, A high–order, adaptive, discontinuous Galerkin finite elemenet method for the Reynolds-Averaged Navier-Stokes equations, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 2008.
  • [31] Michael A. Park, Anisotropic output-based adaptation with tetrahedral cut cells for compressible flows, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 2008.
  • [32] J. Peraire, M. Vahdati, K. Morgan, and O. C. Zienkiewicz, Adaptive remeshing for compressible flow computations, Journal of Computational Physics 72 (1987), 449–466.
  • [33] P.-O. Persson and J. Peraire., Sub-cell shock capturing for discontinuous Galerkin methods, AIAA Paper 2006-112, 2006.
  • [34] W. Rachowicz, D. Pardo, and L. Demkowicz, Fully automatic hp-adaptivity in three dimensions, Tech. Report 04-22, ICES, 2004.
  • [35] R. Rannacher, Adaptive Galerkin finite element methods for partial differential equations, Journal of Computational and Applied Mathematics 128 (2001), 205–233.
  • [36] R. Schneider and P. K. Jimack, Toward anisotropic mesh adaptation based upon sensitivity of a posteriori estimates, Tech. Report 2005.03, University of Leeds, School of Computing, 2005.
  • [37] P. Sol´in and L. Demkowicz, Goal-oriented hp-adaptivity for elliptic problems, Computer Methods in Applied Mechanics and Engineering 193 (2004), 449–468.
  • [38] D. A. Venditti, Grid adaptation for functional outputs of compressible flow simulations, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 2002.
  • [39] D. A. Venditti and D. L. Darmofal, Grid adaptation for functional outputs: application to two-dimensional inviscid flows, Journal of Computational Physics 176 (2002), no. 1, 40–
  • [40] _____, Anisotropic grid adaptation for functional outputs: application to twodimensional viscous flows, Journal of Computational Physics 187 (2003), no. 1, 22–46.
Como citar:

Ceze, Marco; Fidkowski, Krzysztof; "OUTPUT-BASED HP-ADAPTATION APPLIED TO AERODYNAMIC FLOWS", p. 608-627 . In: In Proceedings of the 10th World Congress on Computational Mechanics [= Blucher Mechanical Engineering Proceedings, v. 1, n. 1]. São Paulo: Blucher, 2014.
ISSN 2358-0828, DOI 10.5151/meceng-wccm2012-18038

últimos 30 dias | último ano | desde a publicação