Hyper-reduced order models for parametrized unsteady Navier-Stokes equations on domains with variable shape

  • Niccolò Dal Santo
  • Andrea ManzoniEmail author
Part of the following topical collections:
  1. Model reduction of parametrized Systems


In this work, we set up a new, general, and computationally efficient way to tackle parametrized fluid flows modeled through unsteady Navier-Stokes equations defined on domains with variable shape, when relying on the reduced basis method. We easily describe a domain by flexible boundary parametrizations, and generate domain (and mesh) deformations by means of a solid extension, obtained by solving a harmonic extension or a linear elasticity problem. The proposed procedure is built over a two-stage reduction: (i) first, we construct a reduced basis approximation for the mesh motion problem, irrespective of the fluid flow problem we focus on; (ii) then, we generate a reduced basis approximation of the unsteady Navier-Stokes problem, relying on finite element snapshots evaluated over a set of reduced deformed configuration, and approximating both velocity and pressure fields simultaneously. To deal with unavoidable nonaffine parametric dependencies arising in both the mesh motion and the state problem, we apply a matrix version of the discrete empirical interpolation method, allowing treating a wide range of geometrical deformations in an efficient and purely algebraic way. The same strategy is used to perform hyper-reduction of nonlinear terms. To assess the numerical performances of the proposed technique, we address the solution of parametrized fluid flows where the parameters describe both the shape of the domain and relevant physical features. Complex flow patterns such as the ones appearing in a patient-specific carotid bifurcation are accurately approximated, as well as derived quantities of potential clinical interest.


Reduced basis method Parametrized PDEs Hyper-reduction techniques Navier-Stokes equations Computational fluid dynamics Varying domains 

Mathematics Subject Classification (2010)

65M60 65N12 76M10 


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We acknowledge the Swiss National Supercomputing Centre (CSCS) for providing us the CPU resources under project ID s796.

Funding information

The research of N. Dal Santo has been supported by the Swiss State Secretariat for Education, Research and Innovation (SERI), project No. C14.0068, in the framework of the COST action number TD1307.


  1. 1.
    Baiges, J., Codina, R., Idelsohn, S.: Explicit reduced-order models for the stabilized finite element approximation of the incompressible Navier-Stokes equations. Int. J. Numer. Meth Fluids 72(12), 1219–1243 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Baker, T.J.: Mesh movement and metamorphosis. Eng. Comput. 18(3), 188–198 (2002)CrossRefGoogle Scholar
  3. 3.
    Ballarin, F., Faggiano, E., Ippolito, S., Manzoni, A., Quarteroni, A., Rozza, G., Scrofani, R.: Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization. J. Comput. Phys. 315, 609–628 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ballarin, F., Faggiano, E., Manzoni, A., Quarteroni, A., Rozza, G., Ippolito, S., Antona, C., Scrofani, R.: Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts Biomech. Model Mechan. 16(4), 1373–1399 (2017)zbMATHCrossRefGoogle Scholar
  5. 5.
    Ballarin, F., Manzoni, A., Quarteroni, A., Rozza, G.: Supremizer stabilization of P,OD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations. Int. J. Numer Methods Engng. 102(5), 1136–1161 (2015)zbMATHCrossRefGoogle Scholar
  6. 6.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bergmann, M., Bruneau, C.-H., Iollo, A.: Enablers for robust POD models. J. Comput. Phys. 228(2), 516–538 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bonomi, D., Manzoni, A., Quarteroni, A.: A matrix DEIM technique for model reduction of nonlinear parametrized problems in cardiac mechanics. Comput. Methods Appl. Mech. Engrg. 324, 300–326 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Caiazzo, A., Iliescu, T., John, V., Schyschlowa, S.: A numerical investigation of velocity–pressure reduced order models for incompressible flows. J Comput. Phys. 259, 598–616 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Campbell, I.C., Ries, J., Dhawan, S.S., Quyyumi, A.A., Taylor, W.R., Oshinski, J.N.: Effect of inlet velocity profiles on patient-specific computational fluid dynamics simulations of the carotid bifurcation. J. Biomech. Engng. 134(5), 051001 (2012)CrossRefGoogle Scholar
  11. 11.
    Carlberg, K., Farhat, C., Cortial, J., Amsallem, D.: The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242, 623–647 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Carlberg, K., Tuminaro, R., Boggs, P.: Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics. SIAM J. Sci. Comput. 37(2), B153–B184 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Chaturantabut, S.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Colciago, C.M., Deparis, S., Quarteroni, A.: Comparisons between reduced order models and full 3d models for fluid–structure interaction problems in haemodynamics. J. Comput. Appl. Math. 265, 120–138 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Dal Santo, N., Deparis, S., Manzoni, A., Quarteroni, A.: An algebraic least squares reduced basis method for the solution of parametrized Stokes equations. Technical Report 21.2017 MATHICSE–EPFL (2017)Google Scholar
  16. 16.
    De Sturler, E., Liesen, J.: Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems. part i: Theory. SIAM J. Sci. Comput. 26(5), 1598–1619 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Deparis, S.: Reduced basis error bound computation of parameter-dependent Navier–Stokes equations by the natural norm approach. SIAM J. Numer. Anal. 46 (4), 2039–2067 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Drohmann, M., Haasdonk, B., Ohlberger, M.: Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34(2), A937–A969 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Drohmann, M., Haasdonk, B., Ohlberger, M.: Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34(2), A937–A969 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Díez, P., Zlotnik, S., Huerta, A.: Generalized parametric solutions in Stokes flow. Comput. Methods Appl. Mech. Engrg. 326, 223–240 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Elman, H., Howle, V.E., Shadid, J., Shuttleworth, R., Tuminaro, R.: Block preconditioners based on approximate commutators. SIAM J. Sci Comput. 27 (5), 1651–1668 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Elman, H.C., Forstall, V.: Numerical solution of the parameterized steady-state Navier-Stokes equations using empirical interpolation methods. Comput. Methods Appl. Mech. Engrg. 317, 380–399 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Elman, H.C., Silvester, D.: Fast nonsymmetric iterations and preconditioning for Navier–Stokes equations. SIAM J. Sci. Comput. 17(1), 33–46 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics (2005)Google Scholar
  25. 25.
    Forti, D., Dedè, L.: Semi-implicit bdf time discretization of the Navier–Stokes equations with vms-les modeling in a high performance computing framework. Comput. Fluids 117, 168–182 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Gervasio, P., Saleri, F., Veneziani, A.: Algebraic fractional-step schemes with spectral methods for the incompressible navier–stokes equations. J. Comput. Phys. 214(1), 347–365 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Ghia, U., Ghia, K.N., Shin, C.: High-re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48(3), 387–411 (1982)zbMATHCrossRefGoogle Scholar
  28. 28.
    Guerciotti, B., Vergara, C., Azzimonti, L., Forzenigo, L., Buora, A., Biondetti, P., Domanin, M.: Computational study of the fluid-dynamics in carotids before and after endarterectomy. J. Biomech. 49(1), 26–38 (2016)CrossRefGoogle Scholar
  29. 29.
    Gunzburger, M.D., Peterson, J.S., Shadid, J.N.: Reducer-order modeling of time-dependent PDEs, with multiple parameters in the boundary data. Comput. Methods Appl. Mech. Engrg. 196, 1030–1047 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Helenbrook, B.: Mesh deformation using the biharmonic operator. Int. J. Numer. Methods Engrg. 56(7), 1007–1021 (2003)zbMATHCrossRefGoogle Scholar
  31. 31.
    Ito, K., Ravindran, S.: A reduced-order method for simulation and control of fluid flows. J. Comput. Phys. 143(2), 403–425 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Kay, D., Loghin, D., Wathen, A.: A preconditioner for the steady-state Navier–Stokes equations. SIAM J. Sci Comput. 24(1), 237–256 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Knezevic, D.J., Nguyen, N.-C., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for the parametrized unsteady Boussinesq equations. Math. Mod. Meth. Appl. Sci. 21(07), 1415–1442 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40(2), 492–515 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Lancellotti, R.M., Vergara, C., Valdettaro, L., Bose, S., Quarteroni, A.: Large eddy simulations for blood dynamics in realistic stenotic carotids. Int. J. Numer. Methods Biomed. Engng. 33(11), e2868 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Maday, Y., Nguyen, N.C., Patera, A.T., Pau, S.H.: A general, multipurpose interpolation procedure: the magic points. Commun. Pur. Appl. Anal. 8(1), 383–404 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Manzoni, A.: An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier-Stokes flows. ESAIM Math. Modell. Numer. Anal. 48(4), 1199–1226 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Manzoni, A., Negri, F.: Efficient reduction of PDEs defined on domains with variable shape. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds.) Model Reduction of Parametrized Systems, vol. 17, pp 183–199. Springer, Cham (2017)zbMATHCrossRefGoogle Scholar
  39. 39.
    Manzoni, A., Quarteroni, A., Rozza, G.: Model reduction techniques for fast blood flow simulation in parametrized geometries. Int. J. Numer Methods Biomed. Engng. 28(6-7), 604–625 (2012)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Manzoni, A., Quarteroni, A., Rozza, G.: Shape optimization for viscous flows by reduced basis methods and free-form deformation. Int. J. Numer. Methods Fluids 70(5), 646–670 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    May, D.A., Moresi, L.: Preconditioned iterative methods for Stokes flow problems arising in computational geodynamics. Phys. Earth Planet. In. 171(1), 33–47 (2008)CrossRefGoogle Scholar
  42. 42.
    Negri, F.: Efficient reduction techniques for the simulation and optimization of parametrized systems. Phd thesis EPFL (2015)Google Scholar
  43. 43.
    Negri, F., Manzoni, A., Amsallem, D.: Efficient model reduction of parametrized systems by matrix discrete empirical interpolation. J. Comput. Phys. 303, 431–454 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Negri, F., Manzoni, A., Rozza, G.: Reduced basis approximation of parametrized optimal flow control problems for the S,tokes equations. Comput. Math. Appl. 69, 319–336 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Pagani, S., Manzoni, A., Quarteroni, A.: Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method. Comput. Meth. Appl. Mech. Engrg. 340, 530–558 (2018)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Quarteroni, A., Manzoni, A., Negri, F.: Reduced basis methods for partial differential equations: an introduction. Springer, Berlin (2016)zbMATHCrossRefGoogle Scholar
  47. 47.
    Quarteroni, A., Rozza, G.: Numerical solution of parametrized Navier–Stokes equations by reduced basis methods. Numer. Meth. Part. D.E. 23(4), 923–948 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Rozza, G., Huynh, D., Manzoni, A.: Reduced basis approximation and error bounds for Stokes flows in parametrized geometries: roles of the inf–sup stability constants. Numer. Math. 125(1), 115–152 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2), 461–469 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Segal, A., ur Rehman, M., Vuik, K.: Preconditioners for incompressible Navier-Stokes solvers. Numer. Math. Theory Methods Appl. 3(3), 245–275 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Semaan, R., Kumar, P., Burnazzi, M., Tissot, G., Cordier, L., Noack, B.: Reduced-order modelling of the flow around a high-lift configuration with unsteady coanda blowing. J Fluid Mech. 800, 72–110 (2016)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Sieger, D., Botsch, M., Menzel, S.: On shape deformation techniques for simulation-based design optimization. In: Perotto, S., Formaggia, L. (eds.) New Challenges in Grid Generation and Adaptivity for Scientific Computing, volume 5 of SEMA SIMAI Springer Series, pp 281–303. Springer International Publishing, Switzerland (2015)zbMATHCrossRefGoogle Scholar
  53. 53.
    Silvester, D., Elman, H.C., Kay, D., Wathen, A.: Efficient preconditioning of the linearized Navier–Stokes equations for incompressible flow. J. Comput. Appl Math. 128(1), 261–279 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Slager, C., Wentzel, J., Gijsen, F., Thury, A., Van der Wal, A., Schaar, J., Serruys, P.: The role of shear stress in the destabilization of vulnerable plaques and related therapeutic implications. Nature Reviews Cardiology 2(9), 456 (2005)Google Scholar
  55. 55.
    Staten, M.L., Owen, S.J., Shontz, S.M., Salinger, A.G., Coffey, T.S.: A comparison of mesh morphing methods for 3d shape optimization. In: Proceedings of the 20th international meshing roundtable, pp. 293–311. Springer (2011)Google Scholar
  56. 56.
    Stein, K., Tezduyar, T., Benney, R.: Mesh moving techniques for fluid-structure interactions with large displacements. J. Appl. Mech. 70(1), 58–63 (2003)zbMATHCrossRefGoogle Scholar
  57. 57.
    Stein, K., Tezduyar, T.E., Benney, R.: Automatic mesh update with the solid-extension mesh moving technique. Comput. Methods Appl. Mech. Engrg. 193 (21–22), 2019–2032 (2004)zbMATHCrossRefGoogle Scholar
  58. 58.
    Tezduyar, T., Behr, M., Mittal, S., Johnson, A.: Computation of unsteady incompressible flows with the stabilized finite element methods: space-time formulations, iterative strategies and massively parallel implementations. In: New Methods in Transient Analysis, volume 246/AMD, pp. 7–24. ASME, New York (1992)Google Scholar
  59. 59.
    Turek, S.: Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approache, vol. 6. Springer Science & Business Media (1999)Google Scholar
  60. 60.
    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. Method Fluids 47(8–9), 773–788 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Vuik, K., Saghir, A., Boerstoel, G.: The Krylov accelerated SIMPLE (R) method for flow problems in industrial furnaces. Int. J. Numer. Methods Fluids 33 (7), 1027–1040 (2000)zbMATHCrossRefGoogle Scholar
  62. 62.
    Washabaugh, K., Zahr, M., Farhat, C.: On the use of discrete nonlinear reduced-order models for the prediction of steady-state flows past parametrically deformed complex geometries. In: 54th AIAA Aerospace Sciences Meeting, pp. 1814 (2016)Google Scholar
  63. 63.
    Wathen, A., Silvester, D.: Fast iterative solution of stabilised stokes systems. part i Using simple diagonal preconditioners. SIAM J. Numer. Anal. 30(3), 630–649 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Weller, J., Lombardi, E., Bergmann, M., Iollo, A.: Numerical methods for low-order modeling of fluid flows based on POD. Int. J. Numer. Methods Fluids 63(2), 249–268 (2010)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Wirtz, D., Sorensen, D., Haasdonk, B.: A posteriori error estimation for DEIM reduced nonlinear dynamical systems. SIAM J. Sci. Comput. 36(2), A311–A338 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Wittum, G.: Multi-grid methods for Stokes and Navier-Stokes equations. Numer. Math. 54(5), 543–563 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Wootton, D.M., Ku, D.N.: Fluid mechanics of vascular systems, diseases, and thrombosis. Ann. Rev. Biomed. Eng. 1(1), 299–329 (1999)CrossRefGoogle Scholar
  68. 68.
    Yano, M.: A space-time petrov–galerkin certified reduced basis method: application to the Boussinesq equations. SIAM J. Sci. Comput. 36(1), A232–A266 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Zimmermann, R., Vendl, A., Görtz, S.: Reduced-order modeling of steady flows subject to aerodynamic constraints. AIAA J. 52(2), 255–266 (2014)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.École Polytechnique Fédérale de Lausanne (EPFL)School of MathematicsLausanneSwitzerland
  2. 2.Department of Mathematics, MOX-Laboratory for Modeling and Scientific ComputingPolitecnico di MilanoMilanoItaly

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