Computational Mechanics

, Volume 55, Issue 6, pp 1181–1190 | Cite as

Goal-oriented model adaptivity for viscous incompressible flows

  • T. M. van Opstal
  • P. T. Bauman
  • S. Prudhomme
  • E. H. van Brummelen
Original Paper


In van Opstal et al. (Comput Mech 50:779–788, 2012) airbag inflation simulations were performed where the flow was approximated by Stokes flow. Inside the intricately folded initial geometry the Stokes assumption is argued to hold. This linearity assumption leads to a boundary-integral representation, the key to bypassing mesh generation and remeshing. It therefore enables very large displacements with near-contact. However, such a coarse assumption cannot hold throughout the domain, where it breaks down one needs to revert to the original model. The present work formalizes this idea. A model adaptive approach is proposed, in which the coarse model (a Stokes boundary-integral equation) is locally replaced by the original high-fidelity model (Navier–Stokes) based on a-posteriori estimates of the error in a quantity of interest. This adaptive modeling framework aims at taking away the burden and heuristics of manually partitioning the domain while providing new insight into the physics. We elucidate how challenges pertaining to model disparity can be addressed. Essentially, the solution in the interior of the coarse model domain is reconstructed as a post-processing step. We furthermore present a two-dimensional numerical experiments to show that the error estimator is reliable.


Finite element Boundary element Adaptivity 


  1. 1.
    Barnett AH (2014) Evaluation of layer potentials close to the boundary for Laplace and Helmholtz problems on analytic planar domains. SIAM J Sci Comput 36(2):A427–A451CrossRefGoogle Scholar
  2. 2.
    Bauman PT, Oden JT, Prudhomme S (2009) Adaptive multiscale modeling of polymeric materials with Arlequin coupling and goals algorithms. Comput Methods Appl Mech Eng 198(5–8):799–818MATHCrossRefGoogle Scholar
  3. 3.
    Braack M, Ern A (2003) A posteriori control of modeling errors and discretization errors. Multiscale Model Simul 1(2):221–238MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Brezzi F, Falk R (1991) Stability of higher-order Hood-Taylor methods. SIAM J Numer Anal 28(3):581–590MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Carstensen C, Funken SA, Stephan EP (1997) On the adaptive coupling of FEM and BEM in 2D elasticity. Numer Math 77:187–221MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Delfour MC, Zolésio J-P (2011) Shapes and geometries: metrics, analysis, differential calculus, and optimization. In: Advances in design and control, 2nd edn, SIAMGoogle Scholar
  7. 7.
    Dörfler W (1996) A convergent adaptive algorithm for Poisson’s equation. SIAM J Numer Anal 30:1106–1124CrossRefGoogle Scholar
  8. 8.
    Ern A, Guermond JL (2004) Applied mathematical sciences. Theory and practice of finite elements. Springer, BerlinCrossRefGoogle Scholar
  9. 9.
    Ern A, Perotto S, Veneziani A (2008) Hierarchical model reduction for advection-diffusion-reaction problems. In: Kunisch Karl, Of Günther, Steinbach Olaf (eds) Numericalmathematics and advanced applications. Springer, Berlin/Heidelberg, pp 703–710Google Scholar
  10. 10.
    Fatone L, Gervasio P, Quarteroni A (2000) Multimodels for incompressible flows. J Math Fluid Mech 2:126–150MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Fatone L, Gervasio P, Quarteroni A (2001) Multimodels for incompressible flows: iterative solutions for the Navier-Stokes/Oseen coupling. Math Model Numer Anal 35(3):549–574MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Geuzaine C, Remacle J-F (2009) Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Methods Eng 79(11):1309–1331MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Hoffman J, Jansson J, De Abreu RV (2011) Adaptive modeling of turbulent flow with residual based turbulent kinetic energy dissipation. Comput Methods Appl Mech Eng 200(37–40):2758–2767. Special issue on modeling error estimation and adaptive modelingGoogle Scholar
  14. 14.
    Ladyzhenskaya OA (1963) The mathematical theory of viscous incompressible flows. Mathematics and its applications. Gordon and Breach, New York (Revised english edition)Google Scholar
  15. 15.
    McLean WCH (2000) Strongly elliptic systems and boundary integral equations. Cambridge University Press, CambridgeMATHGoogle Scholar
  16. 16.
    Moghadam ME, Bazilevs Y, Hsia T-Y, Vignon-Clementel IE, Marsden AL, MOCHA (2011) A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulation. Comput Mech 48:277–291MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Oden JT, Prudhomme S (2002) Estimation of modeling error in computational mechanics. J Comput Phys 182:496–515MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Oden JT, Vemaganti KS (2000) Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. J Comput Phys 164:22–47MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    van Opstal TM, van Brummelen EH, van Zwieten GJ (2014) A finite-element/boundary-element method for three-dimensional, large-displacement fluid-structure-interaction. Comput Methods Appl Mech Eng 284:637–663CrossRefGoogle Scholar
  20. 20.
    Perotto S (2006) Adaptive modeling for free-surface flows. ESAIM Math Model Numer Anal 40(3):469–499Google Scholar
  21. 21.
    Prudhomme S, Ben H (2008) Dhia, P.T. Bauman, N. Elkhodja, and J.T. Oden. Computational analysis of modeling error for the coupling of particle and continuum models by the arlequin method. Comput Methods Appl Mech Eng 197(41–42):3399–3409Google Scholar
  22. 22.
    Prudhomme S, Oden JT (2002) Computable error estimators and adaptive techniques for fluid flow problems. In: Lecture notes in computational science and engineering, vol 25. Springer-Verlag, pp 207–268Google Scholar
  23. 23.
    Schwab C (1998) \(p\)- and \(hp\)- finite element methods. Oxford Science Publications, OxfordGoogle Scholar
  24. 24.
    Stein E, Rüter M (2004) Encyclopedia of computational mechanics, volume 2 structures, chapter 2 finite element methods for elasticity with error-controlled discretization and model adaptivity, Wiley, pp 5–58Google Scholar
  25. 25.
    Stein E, Rüter M, Ohnimus S (2011) Implicit upper bound error estimates for combined expansive model and discretization adaptivity. Comput Methods Appl Mech Eng 200(37–40):2626–2638. Special issue on modeling error estimation and adaptive modelingGoogle Scholar
  26. 26.
    Steinbach O (2008) Numerical approximation methods for elliptic boundary value problems: finite and boundary elements. Springer, BerlinCrossRefGoogle Scholar
  27. 27.
    van Brummelen EH, van der Zee KG, Garg VV, Prudhomme S (2012) Flux evaluation in primal and dual boundary-coupled problems. J Appl Mech 79(1):010904CrossRefGoogle Scholar
  28. 28.
    van der Zee KG, van Brummelen EH, Akkerman I, de Borst R (2011) Goal-oriented error estimation and adaptivity for fluid-structure interaction using exact linearized adjoints. Comput Methods Appl Mech Eng 200(37–40):2738–2757. Special issue on modeling error estimation and adaptive modelingGoogle Scholar
  29. 29.
    van Opstal TM, van Brummelen EH (2013) A potential boundary element method for large-displacement fluid–structure interaction. Comput Methods Appl Mech Eng 193:1–23MathSciNetGoogle Scholar
  30. 30.
    van Opstal TM, van Brummelen EH, de Borst R, Lewis MR (2012) A finite-element/boundary-element method for large-displacement fluid–structure interaction. Comput Mech 50(6):779–788MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Wendland WL, Stephan EP, Hsiao GC (1979) On the integral equation methods for plane mixed boundary value problems for the Laplacian. Math Methods Appl Sci 1:265–321MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Zienkiewicz OC, Kelly DW, Bettess P (1979) Marriage à la mode: the best of both worlds (Finite elements and boundary integrals), Wiley, pp 59–80Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • T. M. van Opstal
    • 1
  • P. T. Bauman
    • 2
  • S. Prudhomme
    • 3
  • E. H. van Brummelen
    • 4
  1. 1.Department of Mathematical SciencesNorwegian University of Science and Technology (NTNU)TrondheimNorway
  2. 2.Mechanical and Aerospace Engineering, Computational and Data-Enabled Science and Engineering, University at BuffaloState University of New YorkBuffaloUSA
  3. 3.Département de Mathématiques et de Génie IndustrielÉcole Polytechnique de MontréalMontréalCanada
  4. 4.Department of Mechanical Engineering, Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenNetherlands

Personalised recommendations