Anisotropic Mesh Adaptation

Many physical problems in CFD are characterized by solutions exhibiting directional features.
Navier–Stokes, Stokes and Euler equations are typical examples where boundary, internal layers or shocks may develop. In these situations, the effectiveness of finite element procedures can be improved if the mesh is suitably oriented. However, standard a priori and a posteriori procedures do not provide enough information to control mesh orientation. Some anisotropic techniques based on heuristic approaches have been devised in the past. A typical methodology consists of estimating the Hessian and/or the gradient of the numerical solution and then using this information to drive the mesh
adaption procedure. Although the results are sometimes impressive, these techniques yet lack a rigorous link with a bound of the discretization error.

More rigorous approaches using theoretically based anisotropic adaptivity have been developed in the latest ten years. In particular, the approach that we employ exploits the spectral properties of the standard affine map between the reference triangle and the physical one, thus allowing us to determine the size, shape, and orientation of the mesh elements. These features are automatically identified in order to control a given Sobolev norm of the discretization error or a physically meaningful quantity (lift and drag around bodies in external flows, mean or local values) such that either the number of mesh elements or the accuracy of the approximation is fixed. The final adapted mesh is obtained through a metric-based adaptive procedure.

More recently, we have extended this approach to mesh adaptation on surfaces.

Keywords :
anisotropic mesh adaptation, interpolation error analysis, a posteriori error estimation, residual-based error estimators, goal oriented analysis, recovery based error estimators, surface mesh adaptation, space-time mesh adaptation, mesh adaptation for optimal control problems, minimization of non-smooth and non-convex energy functionals.

AMS classification :
65K10, 65N15, 65N50, 65N30, 76M10, 35K20, 65M50, 65M60, 65D05.

MOX Staff :

  • Simona Perotto
  • Stefano Micheletti
  • Luca Formaggia
  • Marianna Signorini

Collaborations :

  • M. Artina, M. Fornasier (TUM, München)
  • T. Coupez (Ecole Centrale de Nantes)
  • F. Dassi, H. Si (WIAS Berlin)
  • P.E. Farrell (University of Oxford)
  • G. Gorman (Imperial College London)
  • A. Guadagnini, G. Porta (D.I.I.A.R., Politecnico di Milano)
  • C. Maurini (Pierre and Marie Curie University – Paris 6)

Numerical Gallery :

 Projects :

  • PRIN 2010-2011 ”Innovative Methods for Water Resources under Hydro-Climatic Uncertainty Scenarios

Minisymposia and Conferences :

  • TETRAHEDRON IV, The Fourth Tetrahedron Workshop on Grid Generation for Numerical Computations. Verbania, 1-3 July 2013.
  • Minisymposium “Mesh & Adaptivity” at ADMOS 2015 (organized by S. Perotto, T. Coupez).

Publications :

  • Peer-reviewed Journals
  1. L. Formaggia, S. Perotto. New anisotropic a priori error estimates. Numer. Math., 89 (2001), 641-667.
  2. L. Formaggia, S. Perotto, P. Zunino.  An anisotropic a-posteriori error estimate for a convection-diffusion problem.  Comput. Visual. Sci., 4 (2001), no.2,  99-104.
  3. L. Formaggia, S. Perotto.  Anisotropic error estimates for elliptic problems. Numer. Math., 94 (2003), 67-92.
  4. S. Micheletti, S. Perotto, M. Picasso. Stabilized finite elements on anisotropic meshes: a priori error estimates for the  advection-diffusion and the Stokes problems. SIAM J. Numer. Anal., 41 (2003), no.3,  1131-1162.
  5. L. Formaggia, S. Micheletti, S. Perotto. Anisotropic mesh adaptation in Computational Fluid Dynamics: application to the advection-diffusion-reaction and the Stokes problems. Appl. Numer. Math., 51 (2004), no.4, 511-533.
  6. S. Micheletti, S. Perotto. Reliability and efficiency of an anisotropic Zienkiewicz-Zhu error estimator. Comput. Methods Appl. Mech. Engrg., 195 (2006), no.9-12, 799-835.
  7. L. Dedè, S. Micheletti, S. Perotto. Anisotropic error control for environmental applications. Appl. Numer. Math., 58 (2008), no.9, 1320-1339.
  8. S. Micheletti, S. Perotto. Output functional control for nonlinear equations driven by anisotropic mesh adaption. The Navier-Stokes equations.
    SIAM J. Sci. Comput., 30 (2008), no.6, 2817-2854.
  9. S. Micheletti, S. Perotto. Anisotropic mesh adaption for time-dependent problems. Int. J. Numer. Meth. Fluids, 58 (2008), 1009-1015.
  10. S. Micheletti, S. Perotto. Space-time adaptation for purely diffusive problems in an anisotropic framework. Int. J. Numer. Anal. Model., 7 (2010), no.1, 125-155.
  11. P.E. Farrell, S. Micheletti, S. Perotto. A recovery-based error estimator for anisotropic mesh adaptation in CFD. Bol. Soc. Esp. Mat. Apl., 50 (2010), 115-138.
  12. P.E. Farrell, S. Micheletti, S. Perotto. An anisotropic Zienkiewicz-Zhu type error estimator for 3D applications. Int. J. Numer. Methods Engrg., 85 (2011), 671-692.
  13. S. Micheletti, S. Perotto. The effect of anisotropic mesh adaptation on PDE-constrained optimal control problems. SIAM J. Control. Optim., 49 (2011), no.4, 1793-1828.
  14. G.M. Porta, S. Perotto, F. Ballio. Anisotropic mesh adaptation driven by a recovery based error estimator for shallow water flow modeling. Int. J. Numer. Meth. Fluids, 70 (2012), no.3, 269-299.
  15. G.M. Porta, S. Perotto, F. Ballio. A space-time adaptation scheme for unsteady shallow water problems. Mathematics and Computers in Simulation, 82 (2012), 2929-2950.
  • Peer-reviewed Proceedings
  1. S. Micheletti, S. Perotto. An anisotropic recovery-based a posteriori error estimator. In Numerical Mathematics and Advanced Applications, Springer Verlag Italia, F. Brezzi, A. Buffa, S. Corsaro, A. Murli Eds. (2003), 731-741.
  2. S. Micheletti, S. Perotto. Anisotropic mesh adaptivity in CFD. In Adaptive Mesh Refinement – Theory and Applications. Series: Lect. Notes Comput. Sci. Eng., Vol. 41, Springer-Verlag, T. Plewa, T. Linde, V.G. Weirs Eds. (2005), 171-182.
  3. S. Micheletti, S. Perotto. Anisotropic mesh adaptivity via a dual-based a posteriori error estimation for semiconductors. In Scientific Computing in Electrical Engineering. Series: Mathematics in Industry, Vol. 9, Springer-Verlag, Berlin, A.M. Anile, G. Alì, G. Mascali Eds. (2006), 369-375.
  4. S. Micheletti, S. Perotto. Space-time adaption for advection-diffusion-reaction problems on anisotropic meshes. In Numerical Mathematics and Advanced Applications, Springer-Verlag, Berlin Heidelberg, K. Kunisch, G. Of, O. Steinbach Eds. (2008), 49-56.
  5. S. Micheletti, S. Perotto. Anisotropic adaptation via a Zienkiewicz-Zhu error estimator for 2D elliptic problems. In Numerical Mathematics and Advanced Applications, Springer-Verlag, Berlin Heidelberg, G. Kreiss, P. Lotstedt, A. Malqvist, M. Neytcheva Eds. (2010), 645-653.
  6. S. Micheletti, S. Perotto. Anisotropic recovery-based a posteriori error estimators for advection-diffusion-reaction problems. In Numerical Mathematics and Advanced Applications, Springer-Verlag, Berlin Heidelberg, A. Cangiani, R.L. Davidchack, E. Georgoulis, A.N. Gorban, J. Levesley, M.V. Tretyakov Eds (2013), 43-51.
  7. M. Artina, M. Fornasier, S. Micheletti, S.Perotto. Anisotropic adaptive meshes for brittle fractures: parameter sensitivity. To appear in Numerical Mathematics and Advanced Applications, Springer-Verlag, Berlin Heidelberg, A. Abdulle, S. Deparis, D. Kressner, F. Nobile, M. Picasso Eds. (2014).
  • Contributed Books
  1. B. Esfandiar, G.M. Porta, S. Perotto, A. Guadagnini. Anisotropic mesh and time step adaptivity for solute transport modeling in porous media. To appear in New Challenges in Grid Generation and Adaptivity for Scientific Computing. Series: SEMA SIMAI Springer, Springer Milano, S. Perotto, L. Formaggia Eds. (2015).
  2. M. Artina, M. Fornasier, S. Micheletti, S.Perotto. The benefits of anisotropic mesh adaptation for brittle fractures under plane-strain conditions. To appear in New Challenges in Grid Generation and Adaptivity for Scientific Computing. Series: SEMA SIMAI Springer, Springer Milano, S. Perotto, L. Formaggia Eds. (2015).

 

 

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