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Journal of Scientific Computing

, Volume 74, Issue 1, pp 197–219 | Cite as

Certified Reduced Basis Approximation for the Coupling of Viscous and Inviscid Parametrized Flow Models

  • Immanuel Martini
  • Bernard Haasdonk
  • Gianluigi Rozza
Article
  • 189 Downloads

Abstract

We present a model order reduction approach for parametrized laminar flow problems including viscous boundary layers. The viscous effects are captured by the incompressible Navier–Stokes equations in the vicinity of the boundary layer, whereas a potential flow model is used in the outer region. By this, we provide an accurate model that avoids imposing the Kutta condition for potential flows as well as an expensive numerical solution of a global viscous model. To account for the parametrized nature of the problem, we apply the reduced basis method. The accuracy of the reduced order model is ensured by rapidly computable a posteriori error estimates. The main contributions of this paper are the combination of an offline-online splitting with the domain decomposition approach, reducing both offline and online computational loads and a new kernel interpolation method for the approximation of the stability factor in the online evaluation of the error estimate. The viability of our approach is demonstrated by numerical experiments for the section of a NACA airfoil.

Keywords

Reduced basis method Coupled problems Error estimation Navier–Stokes equations 

Mathematics Subject Classification

65N55 76D09 76G25 

Notes

Acknowledgements

The authors would like to thank the Trans-Domain COST Action TD1307 “European Model Reduction Network” (EU-MORNET) for the grant of a Short Term Scientific Mission (STSM), allowing I. Martini to spend a visiting period at SISSA. We also acknowledge financial support by the German Research Foundation (DFG) within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart and by the Baden-Württemberg Stiftung gGmbH. G. Rozza acknowledges the NOFYSAS program: New Opportunities for Young Scientists at SISSA, Trieste, Italy and the INDAM-GNCS projects 2015–2016.

References

  1. 1.
    Abbott, I., von Doenhoff, A.: Theory of Wing Sections. Dover Publications, New York (1959)Google Scholar
  2. 2.
    Barrault, M., Maday, Y., Nguyen, N., Patera, A.: An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Acad. Sci. Ser. I(339), 667–672 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Benner, P., Feng, L.: Model order reduction for coupled problems. Tech. Rep. MPIMD/15-02, Max Planck Institue for Dynamics of Complex Technical Systems, Magdeburg (2015)Google Scholar
  4. 4.
    Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G., Wojtaszczyk, P.: Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43(3), 1457–1472 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brezzi, F., Rappaz, J., Raviart, P.: Finite dimensional approximation of nonlinear problems. Part I: branches of nonsingular solutions. Numer. Math. 36, 1–25 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Buhr, A., Engwer, C., Ohlberger, M., Rave, S.: A numerically stable a posteriori error estimator for reduced basis approximations of elliptic equations. In: Proceedings of the 11th World Congress on Computational Mechanics CIMNE, Barcelona, pp. 4094–4102 (2014)Google Scholar
  7. 7.
    Caloz, G., Rappaz, J.: Numerical analysis for nonlinear and bifurcation problems. In: Ciarlet, P., Lions, J. (eds.) Handbook of Numerical Analysis, pp. 487–637. Elsevier, Amsterdam (1997)Google Scholar
  8. 8.
    Deparis, S., Løvgren, E.: Stabilized reduced basis approximation of incompressible three-dimensional Navier–Stokes equations in parametrized deformed domains. J. Sci. Comput. 50(1), 198–212 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dinh, Q., Glowinski, R., Périaux, J., Terrasson, G.: On the coupling of viscous and inviscid models for incompressible fluid flows via domain decomposition. In: Domain Decomposition Methods for Partial Differential Equations, pp. 350–369. SIAM, Philadelphia (1988)Google Scholar
  10. 10.
    Discacciati, M., Quarteroni, A.: Navier–Stokes/Darcy coupling: modeling, analysis, and numerical approximation. Rev. Mat. Complut. 22(2), 315–426 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Eftang, J., Patera, A.: Port reduction in parametrized component static condensation: approximation and a posteriori error estimation. Int. J. Numer. Methods Eng. 96(5), 269–302 (2013). doi: 10.1002/nme.4543 MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gerner, A., Veroy, K.: Certified reduced basis methods for parametrized saddle point problems. SIAM J. Sci. Comput. 34(5), A2812–A2836 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Giraud, L., Langou, J., Rozloznik, M.: The loss of orthogonality in the Gram-Schmidt orthogonalization process. Comput. Math. Appl. 50, 1069–1075 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  15. 15.
    Grepl, M., Patera, A.: A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. Math. Model. Numer. Anal. 39, 157–181 (2005)CrossRefzbMATHGoogle Scholar
  16. 16.
    Haasdonk, B., Ohlberger, M.: Adaptive basis enrichment for the reduced basis method applied to finite volume schemes. In: Proceedings of 5th International Symposium on Finite Volumes for Complex Applications, pp. 471–478 (2008)Google Scholar
  17. 17.
    Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized linear evolution equations. Math. Model. Numer. Anal. 42, 277–302 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hesthaven, J., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics. Springer, Berlin (2016)CrossRefzbMATHGoogle Scholar
  19. 19.
    Huynh, D., Knezevic, D., Chen, Y., Hesthaven, J., Patera, A.: A natural-norm successive constraint method for inf-sup lower bounds. Comput. Methods Appl. Mech. Eng. 199, 1963–1975 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Huynh, D., Knezevic, D., Patera, A.: A static condensation reduced basis element method: approximation and a posteriori error estimation. Math. Model. Numer. Anal. 47(1), 213–251 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Huynh, D., Rozza, G., Sen, S., Patera, A.: A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. Comptes Rendus Acad. Sci. Ser. I(345), 473–478 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Iapichino, L., Quarteroni, A., Rozza, G.: Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries. Comput. Math. Appl. 71(1), 408–430 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Iapichino, L., Quarteroni, A., Rozza, G.: A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks. Comput. Methods Appl. Mech. Eng. 221–222, 62–82 (2012)MathSciNetzbMATHGoogle Scholar
  24. 24.
    John, V.: Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier–Stokes equations. Int. J. Numer. Methods Fluids 40, 775–798 (2002). doi: 10.1002/fld.377 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Løvgren, A., Maday, Y., Rønquist, E.: A reduced basis element method for the steady Stokes problem. Math. Model. Numer. Anal. 40, 529–552 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Maday, Y., Rønquist, E.: A reduced-basis element method. J. Sci. Comput. 17, 447–459 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Maday, Y., Rønquist, E.: The reduced basis element method: application to a thermal fin problem. J. Sci. Comput. 26, 240–258 (2004)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Maier, I., Haasdonk, B.: A Dirichlet–Neumann reduced basis method for homogeneous domain decomposition problems. Appl. Numer. Math. 78, 31–48 (2014). doi: 10.1016/j.apnum.2013.12.001 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Manzoni, A.: Reduced models for optimal control, shape optimization and inverse problems in haemodynamics. Ph.D. thesis, EPFL, Lausanne (2012)Google Scholar
  30. 30.
    Manzoni, A.: An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier–Stokes flows. Math. Model. Numer. Anal. 48(4), 1199–1226 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Manzoni, A., Negri, F.: Rigorous and heuristic strategies for the approximation of stability factors in nonlinear parametrized PDEs. Adv. Comput. Math. 41(5), 1255–1288 (2015). doi: 10.1007/s10444-015-9413-4 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Manzoni, A., Salmoiraghi, F., Heltai, L.: Reduced basis isogeometric methods (RB-IGA) for the real-time simultion of potential flows about parametrized NACA airfoils. Comput. Methods Appl. Mech. Eng. 284, 1147–1180 (2015)CrossRefGoogle Scholar
  33. 33.
    Martini, I., Haasdonk, B.: Output error bounds for the Dirichlet–Neumann reduced basis method. In: Numerical Mathematics and Advanced Applications-ENUMATH 2013, Lecture Notes in Computational Science and Engineering, vol. 103, pp. 437–445 (2015). doi: 10.1007/978-3-319-10705-9_43
  34. 34.
    Martini, I., Rozza, G., Haasdonk, B.: Reduced basis approximation and a-posteriori error estimation for the coupled Stokes–Darcy system. Adv. Comput. Math. 41(5), 1131–1157 (2015). doi: 10.1007/s10444-014-9396-6 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Panton, R.: Incompressible Flow. Wiley, Hoboken (1996)zbMATHGoogle Scholar
  36. 36.
    Patera, A., Rozza, G.: Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. MIT Pappalardo Graduate Monographs in Mechanical Engineering (2007). http://augustine.mit.edu
  37. 37.
    Pitton, G., Rozza, G.: On the application of reduced basis methods to bifurcation problems in incompressible fluid dynamics. J. Sci. Comput. (2017). doi: 10.1007/s10915-017-0419-6
  38. 38.
    Quarteroni, A., Rozza, G., Manzoni, A.: Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1, 3 (2011). doi: 10.1186/2190-5983-1-3 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Quarteroni, A., Sacchi Landriani, G., Valli, A.: Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements. Numer. Math. 59, 831–859 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, Oxford (1999)zbMATHGoogle Scholar
  41. 41.
    Rozza, G., Huynh, D., Manzoni, A.: Reduced basis approximation and a posteriori error estimation for Stokes flow in parametrized geometries: roles of the inf-sup stability constants. Numer. Math. 125, 115–152 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Rozza, G., Huynh, D., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15, 229–275 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Rozza, G., Veroy, K.: On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Eng. 196, 1244–1260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Schenk, K., Hebeker, F.: Coupling of two-dimensional viscous and inviscid incompressible stokes equations. In: Hebeker, F., Rannacher, R., Wittum, G. (eds.) Numerical Methods for the Navier–Stokes Equations, pp. 239–248. Springer Fachmedien Wiesbaden, Wiesbaden (1993)Google Scholar
  45. 45.
    Smetana, K.: A new certification framework for the port reduced static condensation reduced basis element method. Comput. Methods Appl. Mech. Eng. 283, 352–383 (2015)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing, Providence (2001)Google Scholar
  47. 47.
    Vallaghé, S., Patera, A.: The static condensation reduced basis element method for a mixed-mean conjugate heat exchanger model. SIAM J. Sci. Comput. 36(3), B294–B320 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Veroy, K., Patera, A.: Certified real-time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Methods Fluids 47, 773–788 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Veroy, K., Prud’homme, C., Rovas, D., Patera, A.: A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: Proceedings of 16th AIAA Computational Fluid Dynamics Conference, p. 3847 (2003)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Immanuel Martini
    • 1
  • Bernard Haasdonk
    • 1
  • Gianluigi Rozza
    • 2
  1. 1.Institute for Applied Analysis and Numerical SimulationUniversity of StuttgartStuttgartGermany
  2. 2.SISSA MathLabInternational School for Advanced StudiesTriesteItaly

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