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Adjoint-Based Functional Correction for Unstructured Mesh Finite Volume Methods

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Abstract

The goal of this paper is to demonstrate the use of adjoint-based functional correction and mesh adaptation for aerodynamic flows with an unstructured mesh finite volume solver. A key feature of our approach is that all calculations are performed on a single mesh, unlike other error correction and mesh adaptation schemes. As using the original p-truncation error estimate is not helpful in improving the functional, we use a smoothed estimate of the truncation error to correct the functional for both inviscid and viscous flows. The correction term is based on the smoothed truncation error and the adjoint solution, with both the continuous and discrete adjoints. The same correction term is used as an adaptation indicator for goal-based mesh adaptation as well. Our results show the effectiveness of our method in improving the convergence rate for test cases of interest in computational aerodynamics.

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Notes

  1. Discretization error is the difference between the exact solution and the discrete solution.

Abbreviations

A:

Jacobian matrix

f:

Source term of the primal problem

g:

Source term of the adjoint problem

J:

Output functional

\(\alpha \) :

Angle of attack

\( Ma \) :

Mach number

\( Re \) :

Reynolds number

\( \tau \) :

Visous stress term

References

  1. Becker, R., Rannacher, R.: Weighted a posteriori error control in finite element methods. In: Proceedings of ENUMATH-97, Heidelberg (1998)

  2. Larson, M.G., Barth, T.J.: A posteriori error estimation for discontinuous Galerkin approximations of hyperbolic systems. NAS technical report, number NAS-99-010 (1999)

  3. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, Hoboken (2000)

    Book  MATH  Google Scholar 

  4. Machiels, L., Peraire, J., Patera, A.T.: A posteriori finite element output bounds for the incompresible Navier–Stokes equations: application to a natural convection problem. J. Comput. Phys. 172, 401–425 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Pierce, N., Giles, M.: Adjoint and defect error bounding and correction for functional estimates. J. Comput. Phys. 200(2), 769–794 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Pierce, N., Giles, M.: Adjoint recovery of superconvergent functionals from PDE approximations. SIAM Rev. 42(2), 247–264 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Venditti, D.A., Darmofal, D.L.: Grid adaptation for functional outputs: application to two-dimensional inviscid flows. J. Comput. Phys. 175(1), 40–69 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Venditti, D.A., Darmofal, D.L.: Grid adaptation for functional outputs: application to two-dimensional viscous flows. J. Comput. Phys. 187, 22–46 (2003)

    Article  MATH  Google Scholar 

  9. Venditti, D.A., Darmofal, D.L.: Adjoint error estimation and grid adaptation for functional outputs: application to quasi-one-dimensional flow. J. Comput. Phys. 164(1), 204–227 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ollivier-Gooch, C.F., Van Altena, M.: A high-order accurate unstructured mesh finite-volume scheme for the advection–diffusion equation. J. Comput. Phys. 181(2), 729–752 (2002)

    Article  MATH  Google Scholar 

  11. Jalali, A., Ollivier-Gooch, C.: Accuracy assessment of finite volume discretizations of diffusive fluxes on unstructured meshes. In: Proceedings of the 50th Aerospace Sciences Meeting, AIAA Paper 2012-0608. American Institute of Aeronautics and Astronautics (2012)

  12. Diskin, B., Thomas, J.L.: Accuracy analysis for mixed-element finite-volume discretization schemes. Technical report 2007-08, National Institute of Aerospace (2007)

  13. Diskin, B., Thomas, J.L., Nielsen, E.J., Nishikawa, H., White, J.A.: Comparison of node-centered and cell-centered unstructured finite-volume discretizations. Part I: viscous fluxes. In: Proceedings of the 47th Aerospace Sciences Meeting, AIAA Paper 2009-597 (2009)

  14. Sharbatdar, M., Ollivier-Gooch, C.: Eigenanalysis of truncation and discretization error on unstructured meshes. In: 21st AIAA Computational Fluid Dynamics Conference, AIAA Paper 2013-3089. American Institute of Aeronautics and Astronautics (2013)

  15. Sharbatdar, M., Jalali, A., Ollivier-Gooch, C.: Smoothed truncation error in functional error estimation and correction using adjoint methods in an unstructured finite volume method. Comput Fluids 140, 406–421 (2016)

    Article  MathSciNet  Google Scholar 

  16. Baker, T.J.: Mesh adaptation strategies for problems in fluid dynamics. Finite Elem. Anal. Des. 25, 243–273 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Habashi, W.G., Dompierre, J., Bourgault, Y., Ait-Ali-Yahia, D., Fortin, M., Vallet, M.G.: Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. Part I: general principles. Int. J. Numer. Methods Fluids 32(6), 725–744 (2000)

    Article  MATH  Google Scholar 

  18. Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineerng analysis. Int. J. Numer. Methods Eng. 24, 337–357 (1987)

    Article  MATH  Google Scholar 

  19. Sharbatdar, M, Ollivier-Gooch, C.: Anisotropic mesh adaptation: recovering quasi-structured meshes. In: Proceedings of the 51st Aerospace Science Meeting. AIAA Paper 2013-0149 (2013)

  20. Roe, P.L.: Characteristic-based schemes for the Euler equations. Ann. Rev. Fluid Mech. 18, 337–365 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nishikawa, H.: Beyond interface gradient: a general principle for constructing diffusion scheme. In: Proceedings of the Fortieth AIAA Fluid Dynamics Conference and Exhibit. AIAA Paper 2010-5093 (2010)

  22. Jalali, A., Sharbatdar, M., Ollivier-Gooch, C.: Accuracy analysis of unstructured finite volume discretization schemes for diffusive fluxes. Comput. Fluids 101, 220–232 (2014)

    Article  MathSciNet  Google Scholar 

  23. Michalak, C., Ollivier-Gooch, C.: Globalized matrix-explicit Newton-GMRES for the high-order accurate solution of the Euler equations. Comput. Fluids 39, 1156–1167 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nejat, A.: A Higher-Order Accurate Unstructured Finite Volume Newton–Krylov Algorithm for Inviscid Compressible Flows. PhD Thesis, The University of British Columbia (2007)

  25. Ollivier-Gooch, C., Nejat, A., Michalak, C.: On obtaining and verifying high-order finite-volume solutions to the Euler equations on unstructured meshes. AIAA J. 47(9), 2105–2120 (2009)

    Article  Google Scholar 

  26. Fidkowski, K., Darmofal, D.: Review of output-based error estimation and mesh adaptation in computational fluid dynamics. AIAA J. 49(4), 673–694 (2011)

    Article  Google Scholar 

  27. Hayashi, M., Ceze, M., Volpe, E.: Characteristic-based boundary conditions for the Euler adjoint problem. Int. J. Numer. Methods Fluids 71, 1297–1321 (2013)

    Article  MathSciNet  Google Scholar 

  28. Kautsky, J., Nichols, N.K.: Equidistributing meshes with constraints. SIAM J. Sci. Stat Comput. 1(4), 499–511 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ollivier-Gooch, C.: GRUMMP version 0.7.0 user’s guide. Technical report, Department of Mechanical Engineering, The University of British Columbia (2015)

  30. Argyris, J.H., Fried, I., Scharpf, D.W.: The tuba family of plate elements for the matrix displacement method. Aeronaut. J. R. Aeronaut. Soc. 72, 701–709 (1968)

    Google Scholar 

  31. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002)

    Book  MATH  Google Scholar 

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Acknowledgements

This work was supported by ANSYS Canada and the Canadian Natural Sciences and Engineering Research Council under Collaborative Research and Development Grant 431183-12.

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Authors and Affiliations

  1. Department of Mechanical Engineering, The University of British Columbia, Vancouver, BC, V6T 1Z4, Canada

    Mahkame Sharbatdar & Carl Ollivier-Gooch

Authors
  1. Mahkame Sharbatdar
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  2. Carl Ollivier-Gooch
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Correspondence to Mahkame Sharbatdar.

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Sharbatdar, M., Ollivier-Gooch, C. Adjoint-Based Functional Correction for Unstructured Mesh Finite Volume Methods. J Sci Comput 76, 1–23 (2018). https://doi.org/10.1007/s10915-017-0611-8

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  • DOI: https://doi.org/10.1007/s10915-017-0611-8

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