Skip to main content
SpringerLink
Log in

Missing boundary data recovery using Nash games: the Stokes system

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

We consider the Cauchy-Stokes problem. We use a new method based on Nash game theory to recover the missing velocity and normal stress on some inaccessible part of the boundary. This method is used with two different approaches. The first one is compared to a control type one. The numerical study attests that both approaches give accurate results. We compare these results with those of the energy-like minimization method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aboulaïch, R., Ben Abda, A., Kallel, M.: A control type method for solving the Cauchy-Stokes problem. Appl. Math. Model. 37, 4295–4304 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aboulaich, R., Habbal, A., Kallel, M., Moakher, M.: A Nash-game approach to joint image restoration and segmentation. Appl. Math. Model. 38(11), 3038–3053 (2014)

    MathSciNet  Google Scholar 

  3. Aboulaïch, R., Habbal, A., Moussaid, N.: Study of different strategies for splitting variables in multidisciplinary topology optimization. Appl. Math. Sci. 8(144), 7153–7174 (2014)

    Google Scholar 

  4. Adams, D.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  5. Alvarez, C., Conca, C., Friz, L., Kavian, O., Ortega, J.H.: Identification of immersed obstacles via boundary measurements. Inverse Probl. 21, 919–925 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arthern, R.J., Gudmundsson, G.H.: Initialization of ice-sheet forecasts viewed as an inverse Robin problem. J. Glaciol. 56, 527–533 (2010)

    Article  Google Scholar 

  7. Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Alternating proximal algorithms for weakly coupled convex minimization problems. Application to dynamical games and PDE’s. J. Convex Anal. 15, 485–506 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Aubin, J.P.: Mathematical Methods of Game and Economic Theory. North-Holland Publishing Co, Amsterdam (1979)

    MATH  Google Scholar 

  9. Babuška, I.: The finite element method with penalty. Math. Comput. 27(122), 221–228 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ballerini, A.: Stable determination of an immersed body in a stationary Stokes fluid. Inverse Probl. 26, 125015 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bastay, G., Johansson, T., Kozlov, V.A., Lesnic, D.: An alternating method for the stationary Stokes system. Z.MM, Z. Angew. Math. Mech. 86, 268–280 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ben Abda, A., Ben Saad, I., Hassine, M.: Recovering boundary data: The Cauchy Stokes system. Appl. Math. Model. 37, 1–12 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ben Abda, A., Khayat, F.: Reconstruction of missing boundary conditions from partially overspecified data: The Stokes system. ARIMA J. 23, 79–100 (2016)

    MathSciNet  Google Scholar 

  14. Boulakia, M., Egloffe, A.C., Grandmont, C.: Stability estimates for a Robin coefficient in the two-dimensional Stokes system. Math. Control Related Field 3, 21–49 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Christodoulou, K.N., Scriven, L.E.: The fluid mechanics of slide coating. J. Fluid Mech. 208, 321–354 (1989)

    Article  MATH  Google Scholar 

  16. Denn, M.M.: Issues in viscoelastic fluid mechanics. Annu. Rev. Fluid Mech. 22, 13–32 (1990)

    Article  Google Scholar 

  17. Désidéri, J.A.: Cooperation and competition in multidisciplinary optimization. Comput. Appl. 52, 29–68 (2012)

    Article  MATH  Google Scholar 

  18. Dione, I.: Analyse théorique et numérique des conditions de glissement pour les fluides et les solides par la méthode de pénalisation, Ph. D. Thesis, Université Laval, Québec (2013)

  19. Fernandez, M.A., Gerbeau, J.F., Martin, V.: Numerical simulation of blood flows through a porous interface. ESAIM: Math. Modell. Numer. Anal., 42 (2008)

  20. Fernández-Méndez, S., Huerta, A.: Imposing essential boundary conditions in mesh-free methods. Publ. CIMNE., 236 (2003)

  21. Egloffe, A.C.: Etude de quelques problèmes inverses pour le système de Stokes. Application aux poumons, PhD thesis, Paris VI (2012)

  22. Habbal, A.: A topology Nash game for tumoral antangiogenesis. Struct. Multidisc. Optim. 30, 404–412 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Habbal, A., Kallel, M.: Neumann-Dirichlet Nash strategies for the solution of ellptic Cauchy problems. SIAM J. Control. Optim. 51, 4066–4083 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Habbal, A., Petersson, J., Thellner, M.: Multidisciplinary topology optimization solved as a Nash game. Int. J. Numer. Meth. Engng 61, 949–963 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ingham, D.B.: Steady flow past a rotating cylinder. Comput. Fluids 11(4), 351–366 (1983)

    Article  MATH  Google Scholar 

  26. Johansson, T., Lesnic, D.: A variational conjugate gradient method for determining the fluid velocity of a slow viscous flows from Cauchy data. Appl. Anal. 85(11), 1327–1341 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Johansson, T., Lesnic, D.: Reconstruction of a stationary flow from incomplete boundary data using iterative methods. Eur. J. Appl. Math. 17(5), 651–663 (2007)

    MathSciNet  MATH  Google Scholar 

  28. Johansson, T., Lesnic, D.: An iterative method for the reconstruction of a stationary flow. Numer. Methods Par. Diff. Equa. 23(5), 998–1017 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Karageorghis, A., Lesnic, D.: The pressure-streamfunction MFS formulation for the detection of an obstacle immersed in a two-dimensional Stokes flow. Adv. Appl. Math. Mech. 2(2), 183–199 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Liu, C.S., Young, D.L.: A multiple-scale Pascal polynomial for 2D-Stokes and inverse Cauchy-Stokes problems. J. Comput. Phys. 312, 1–13 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Marin, L., Elliott, L., Ingham, D.B., Lesnic, D.: Boundary element method for the Cauchy problem in linear elasticity. Eng. Anal. Bound. Elem. 25, 783–793 (2001)

    Article  MATH  Google Scholar 

  32. Maury, B: Méthode des éléments finis en élasticité Ecole Polytechnique (2007)

  33. Maury, B.: Modélisation en mécanique, Online reference, https://www.math.u-psud.fr/~maury/paps/perso/meca.pdf

  34. Mittal, S., Kumar, B.: Flow past a rotating cylinder. J. Fluid Mech. 476, 303–334 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Neto, C., Evans, D.R., Bonaccurso, E., Butt, H.J., Craig, V.S.: Boundary slip in Newtonian liquids: A review of experimental studies. Rep. Prog. Phys. 68, 2859–2897 (2005)

    Article  Google Scholar 

  36. Pironneau, O., Hecht, F., Hyaric, A., Ohtsuka, K.: FreeFEM, http://www.freefem.org (2006)

  37. Quarteroni, A., Veneziani, A.: Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations. Multiscale Model. Simul. 1, 173–95 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  38. Tang, Z., Désidéri, J.A., Périaux, J.: Multi-criterion aerodynamic shape-design optimization and inverse problems using control theory and Nash games. J. Optim. Theory Appl. 135, 599–622 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zeb, A., Elliott, L., Ingham, D.B., Lesnic, D.: Boundary element two-dimensional solution of an inverse Stokes problem. Eng. Anal. Bound. Elem. 24(1), 75–88 (2000)

    Article  MATH  Google Scholar 

  40. Zeb, A., Elliott, L., Ingham, D.B., Lesnic, D.: An inverse Stokes problem using interior pressure data. Eng. Anal. Bound. Elem. 26(9), 739–745 (2002)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LAMSIN-ENIT-BP 37, 1002, Tunis, le Belvédère, Tunisia

    A. Ben Abda & F. Khayat

Authors
  1. A. Ben Abda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Khayat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Khayat.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ben Abda, A., Khayat, F. Missing boundary data recovery using Nash games: the Stokes system. Numer Algor 78, 777–803 (2018). https://doi.org/10.1007/s11075-017-0400-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0400-3

Keywords

Search

Navigation