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A recovery-based error estimator for anisotropic mesh adaptation in CFD

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Abstract

We provide a unifying framework that generalizes the 2D and 3D settings proposed in [32] and [17], respectively. In these two works we propose a gradient recovery type a posteriori error estimator for finite element approximations on anisotropic meshes. The novelty is the inclusion of the geometrical features of the computational mesh (size, shape and orientation) in the estimator itself. Moreover, we preserve the good properties of recovery based error estimators, in particular their computational cheapness and ease of implementation. A metric-based optimization procedure, relying on the estimator, drives the anisotropic adaptation of the mesh. The focus of this work then moves to a goal-oriented framework. In particular, we extend the idea proposed in [32, 17] to the control of a goal functional. The preliminary results are promising, since it is shown numerically to yield quasi-optimal triangulations with respect to the error-vs-number of elements behaviour.

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References

  1. M.T. Ainsworth and J.T. Oden. A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York, 2000.

    Book  MATH  Google Scholar 

  2. R.C. Almeida, R.A. Feijóo, A.C. Galeão, C. Padra, and R.S. Silva. Adaptive finite element computational fluid dynamics using an anisotropic error estimator. Comput. Methods Appl. Mech. Engrg., 182(3–4):379–400, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  3. T. Apel. Anisotropic Finite Elements: Local Estimates and Applications. Advances in Numerical Mathematics. Teubner: Stuttgart, 1999.

    Google Scholar 

  4. B.F. Armaly, F. Durst, J.C.F. Pereira, and B. Schönung. Experimental and theoretical investigation of backward-facing step flow. J. Fluid Mech., 127:473–496, 1983.

    Article  Google Scholar 

  5. R. Becker and R. Rannacher. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica, 10:1–102, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  6. C.L. Bottasso, G. Maisano, S. Micheletti, and S. Perotto. On some new recovery based a posteriori error estimators. Comput. Methods Appl. Mech. Engrg., 195:4794–4815, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Bourgault, M. Picasso, F. Alauzet, and A. Loseille. On the use of anisotropic a posteriori error estimators for the adaptative solution of 3D inviscid compressible flows. Int. J. Numer. Meth. Fluids, 59(1):47–74, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Boussetta, T. Coupez, and L. Fourment. Adaptive remeshing based on a posteriori error estimation for forging simulation. Comput. Methods Appl. Mech. Engrg., 195(48–49):6626–6645, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  9. J.C. Bruch. Free surface seepage problems solved using: (a) the Zienkiewicz-Zhu error estimation procedure; and (b) a parallel computer. Pitman Res. Notes Math. Ser., 282:98–104, 1993.

    Google Scholar 

  10. A.S. Candy. Subgrid scale modelling of transport processes. PhD thesis. Imperial College London, 2008.

  11. C. Carstensen. All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable. Math. Comp., 73:1153–1165, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  12. M.J. Castro-Diaz, F. Hecht, B. Mohammadi, and O. Pironneau. Anisotropic unstructured mesh adaptation for flow simulations. Int. J. Numer. Meth. Fluids, 25(4):475–491, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  13. Ph. Clément. Approximation by finite element functions using local regularization. RAIRO Anal. Numér., 2:77–84, 1975.

    Google Scholar 

  14. E.F. D’Azevedo. Optimal triangular mesh generation by coordinate transformation. SIAM J. Sci. Statist. Comput., 12(4):755–786, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Dompierre, M.-G. Vallet, Y. Bourgault, M. Fortin, and W.G. Habashi. Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. III. Unstructured meshes. Int. J. Numer. Meth. Fluids, 39(8):675–702, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  16. K. Eriksson, D. Estep, P. Hansbo, and C. Johnson. Introduction to adaptive methods for differential equations. Acta Numerica, 4:105–158, 1995.

    Article  MathSciNet  Google Scholar 

  17. P.E. Farrell, S. Micheletti, and S. Perotto. An anisotropic Zienkiewicz-Zhu a posteriori error estimator for 3D applications. Technical Report 25/2009, MOX, Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, 2009.

  18. L. Formaggia, S. Micheletti, and S. Perotto. Anisotropic mesh adaptation in computational fluid dynamics: application to the advection-diffusion-reaction and the Stokes problems. Appl. Numer. Math., 51(4):511–533, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  19. L. Formaggia and S. Perotto. New anisotropic a priori error estimates. Numer. Math., 89(4):641–667, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Formaggia and S. Perotto. Anisotropic error estimates for elliptic problems. Numer. Math., 94(1):67–92, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  21. P.J. Frey and F. Alauzet. Anisotropic mesh adaptation for CFD computations. Comput. Methods Appl. Mech. Engrg., 194:5068–5082, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  22. P.L. George and H. Borouchaki. Delaunay Triangulation and Meshing: Application to Finite Elements. Hermes: Paris, 1998.

    MATH  Google Scholar 

  23. M.B. Giles and E. Suli. Adjoint methods for pdes: a posteriori error analysis and postprocessing by duality. Acta Numerica, pages 145–236, 2002.

    Google Scholar 

  24. C. Gruau and T. Coupez. 3D tetrahedral, unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric. Comput. Methods Appl. Mech. Engrg., 194(48–49):4951–4976, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  25. F. Hecht. Freefem++ Manual, Third Edition, Version 3.5, 2009, http://www.freefem.org/ff++/index.htm.

  26. M. Krízek and P. Neittaanmäki. Superconvergence phenomenon in the finite element method arising from averaging gradients. Numer. Math, 45:105–116, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  27. G. Kunert. An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math., 86(3):471–490, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  28. K.L. Lawrence and R.V. Nambiar. The Zienkiewicz-Zhu error estimator for multiple material problems. Commun. Appl. Numer. Methods, 8:273–277, 1992.

    Article  MATH  Google Scholar 

  29. H. Le, P. Moin, and J. Kim. Direct numerical simulation of turbulent flow over a backward-facing step. J. Fluid Mech., 330:349–374, 1997.

    Article  MATH  Google Scholar 

  30. J.L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications, vol. I. Springer-Verlag, Berlin, 1972.

    Book  MATH  Google Scholar 

  31. S. Micheletti and S. Perotto. Output functional control for nonlinear equations driven by anisotropic mesh adaption: the Navier-Stokes equations. SIAM J. Sci. Comput., 30(6):2817–2854, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Micheletti and S. Perotto. Anisotropic adaptation via a Zienkiewicz-Zhu error estimator for 2D elliptic problems. Proceedings of the eighth European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2009, June 29–July 3, Uppsala, Sweden, 2009.

    Google Scholar 

  33. A. Naga and Z. Zhang. A posteriori error estimates based on the polynomial preserving recovery. SIAM J. Numer. Anal., 42:1780–1800, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  34. C.C. Pain, A.P. Umpleby, C.R.E. de Oliveira, and A.J.H. Goddard. Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations. Comput. Methods Appl. Mech. Engrg., 190(29–30):3771–3796, 2001.

    Article  MATH  Google Scholar 

  35. T.P. Pawlak, M.J. Wheeler, and S.M. Yunus. Application of the Zienkiewicz-Zhu error estimator for plate and shell analysis. Int. J. Numer. Meth. Engng, 29:1281–1298, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  36. J. Peraire, M. Vahdati, K. Morgan, and O.C Zienkiewicz. Adaptive remeshing for compressible flow computations. J. Comput. Phys., 72(2):449–466, 1987.

    Article  MATH  Google Scholar 

  37. M.D. Piggott, C.C. Pain, G.J. Gorman, P.W. Power, and A.J.H. Goddard. h, r, and hr adaptivity with applications in numerical ocean modelling. Ocean Model., 10(1–2):95–113, 2005.

    Article  Google Scholar 

  38. R. Rodríguez. Some remarks on Zienkiewicz-Zhu estimator. Numer. Methods Partial Differential Equations, 10:625–635, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  39. L.R. Scott and S. Zhang. Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp., 54:483–493, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  40. R.B. Simpson. Anisotropic mesh transformations and optimal error control. Appl. Numer. Math., 14:183–198, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  41. D.A. Venditti and D.L. Darmofal. Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows. J. Comput. Phys., 187:22–46, 2003.

    Article  MATH  Google Scholar 

  42. N. Yan and A. Zhou. Gradient recovery type a posteriori error estimation for finite element approximations on irregular meshes. Comput. Methods Appl. Mech. Engrg., 190:4289–4299, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  43. O.C. Zienkiewicz and J.Z. Zhu. A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Meth. Engng, 24:337–357, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  44. O.C. Zienkiewicz and J.Z. Zhu. The superconvergent patch recovery and a posteriori error estimates. I: The recovery technique. Int. J. Numer. Meth. Engng, 33:1331–1364, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  45. O.C. Zienkiewicz and J.Z. Zhu. The superconvergent patch recovery and a posteriori error estimates. II: Error estimates and adaptivity. Int. J. Numer. Meth. Engng, 33:1365–1382, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  46. O.C. Zienkiewicz and J.Z. Zhu. The superconvergent patch recovery (SPR) and adaptive finite element refinement. Comput. Methods Appl. Mech. Engrg., 101:207–224, 1992.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, SW7 2AZ, UK

    S. Micheletti & S. Perotto

  2. MOX - Modellistica e Calcolo Scientifico Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, via Bonardi 9, I-20133, Milano, Italy

    P. E. Farrell

Authors
  1. S. Micheletti
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  2. S. Perotto
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  3. P. E. Farrell
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Correspondence to S. Micheletti.

Additional information

Acknowledgement: UK Natural Environment Research Council grant NE/C52101X/1; Imperial College High Performance Computing Service; P.E. Farrell would like to thank AWE for their funding of his research through the Institute of Shock Physics

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Micheletti, S., Perotto, S. & Farrell, P.E. A recovery-based error estimator for anisotropic mesh adaptation in CFD. SeMA 50, 115–137 (2010). https://doi.org/10.1007/BF03322545

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