# Reduced-Order Modelling Strategies for the Finite Element Approximation of the Incompressible Navier-Stokes Equations

• Joan Baiges
• Ramon Codina
• Sergio R. Idelsohn
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 33)

## Abstract

In this chapter we present some Reduced-Order Modelling methods we have developed for the stabilized incompressible Navier-Stokes equations. In the first part of the chapter, we depart from the stabilized finite element approximation of incompressible flow equations and we build an explicit proper-orthogonal decomposition based reduced-order model. To do this, we treat the pressure and all the non-linear terms in an explicit way in the time integration scheme. This is possible due to the fact that the reduced model snapshots and basis functions do already fulfill an incompressibility constraint weakly. This allows a hyper-reduction approach in which only the right-hand-side vector needs to be reconstructed. In the second part of the chapter we present a domain decomposition approach for reduced-order models. The method consists in restricting the reduced-order basis functions to the nodes belonging to each of the subdomains. The method is extended to the particular case in which one of the subdomains is solved by using the high-fidelity, full-order model, while the other ones are solved by using the low-cost, reduced-order equations.

## Keywords

Proper Orthogonal Decomposition Domain Decomposition Finite Element Mesh Proper Orthogonal Decomposition Base Local Basis Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Akhtar I, Borggaard J, Hay A (2010) Shape sensitivity analysis in flow models using a finite-difference approach. Math Probl Eng 1–23:2010
2. 2.
Antil H, Heinkenschloss M, Hoppe RHW, Sorensen DC (2010) Domain decomposition and model reduction for the numerical solution of pde constrained optimization problems with localized optimization variables. Comput Vis Sci 13(6):249–264
3. 3.
Arian E, Fahl M, Sachs EW (2000). Trust-Region proper orthogonal decomposition for flow control. Institute for computers, pp 2000–2101Google Scholar
4. 4.
Astrid P (2004). Reduction of process simulation models: a proper orthogonal decomposition approach. PhD thesis, Department of Electrical Engineering, Eindhoven University of TechnologyGoogle Scholar
5. 5.
Astrid P, Weiland S, Willcox K, Backx T (2008) Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans Autom Control 53:2237–2251
6. 6.
Baiges J, Codina R, Idelsohn S (2013) A domain decomposition strategy for reduced order models. Application to the incompressible Navier-Stokes equations. Comput Meth Appl Mech Eng 267:23–42
7. 7.
Baiges J, Codina R, Idelsohn S (2013) Explicit Reduced Order Models for the stabilized finite element approximation of the incompressible Navier-Stokes equations. Int J Numer Meth Fluids 72:1219–1243
8. 8.
Barrault M, Maday Y, Nguyen NC, Patera AT (2004) An ’empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique 339(9):667–672
9. 9.
Bergmann M, Cordier L, Brancher JP (2007) Drag minimization of the cylinder wake by trust-region proper orthogonal decomposition. Notes on numerical fluid mechanics and multidisciplinary design 95:19Google Scholar
10. 10.
Buffoni M, Telib H, Iollo A (2009) Iterative methods for model reduction by domain decomposition. Comput Fluids 38(6):1160–1167
11. 11.
Bui-Thanh T, Willcox K, Ghattas O (2008) Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J Sci Comput 30(6):3270
12. 12.
Burkardt J, Gunzburger M, Lee H (2006) POD and CVT-based reduced-order modeling of Navier-Stokes flows. Comput Meth Appl Mech Eng 196(1–3):337–355
13. 13.
Carlberg K, Bou-Mosleh C, Farhat C (2011) Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations. Int J Numer Meth Eng 86(2):155–181
14. 14.
Chatterjee A (2000) An introduction to the proper orthogonal decomposition. Curr Sci 78(7):808–817Google Scholar
15. 15.
Chaturantabut S, Sorensen DC (2009). Discrete empirical interpolation for nonlinear model reduction. Technical Report TR09-05, Rice University, Houston, TexasGoogle Scholar
16. 16.
Codina R (2001) A stabilized finite element method for generalized stationary incompressible flows. Comput Meth Appl Mech Eng 190:2681–2706
17. 17.
Drohmann M, Haasdonk B, Ohlberger M (2012) Reduced basis approximation for nonlinear parameterized evolution equations based on empirical operator interpolation. SIAM J Sci Comput 34:937–962
18. 18.
Everson R, Sirovich L (1995) Karhunen-Loève procedure for gappy data. J Opt Soc Am A 12:1657–1664
19. 19.
Galletti B, Bruneau CH, Zannetti L, Iollo A (2004) Low-order modelling of laminar flow regimes past a confined square cylinder. J Fluid Mech 503:161–170
20. 20.
Glaz B, Liu L, Friedmann PP (2010) Reduced-Order nonlinear unsteady aerodynamic modeling using a surrogate-based recurrence framework. AIAA J 48(10):2418–2429
21. 21.
Graham WR, Peraire J, Tang KY (1999) Optimal control of vortex shedding using low-order models. Part i-open-loop model development. Int J Numer Meth Eng 44(7):945–972
22. 22.
Grepl MA, Maday Y, Nguyen NC, Patera AT (2007) Efficient Reduced-Basis treatment of nonaffine and nonlinear partial differential equations. ESAIM. Math Model Numer Anal 41(03):575–605
23. 23.
Holmes P, Lumley JL, Berkooz G (1998) Turbulence, coherent structures. Dynamical systems and symmetry. Cambridge University Press, New York
24. 24.
Jacobs EN, Ward KE, Pinkerton RM (1933). The characteristics of 78 related airfoil sections from tests in the variable-density wind tunnel. NACA report, 460Google Scholar
25. 25.
Kalashnikova I, Barone MF (2011). Stable and efficient galerkin reduced order models for non-linear fluid flow. In: AIAA-2011-3110, 6th AIAA theoretical fluid mechanics conference, HonoluluGoogle Scholar
26. 26.
Kosambi DD (1943) Statistics in function space. J Indian Math Soc 7:76–88
27. 27.
Lassila T, Rozza G (2010) Parametric free-form shape design with PDE models and reduced basis method. Comput Meth Appl Mech Eng 199(23–24):1583–1592
28. 28.
LeGresley PA (2005). Application of proper orthogonal decomposition to design decomposition methods. PhD thesis, Department of Aeronautics and Astronautics, Stanford UniversityGoogle Scholar
29. 29.
Lucia DJ, Beran PS (2003) Projection methods for reduced order models of compressible flows. J Comput Phys 188(1):252–280
30. 30.
Lucia DJ, King PI, Beran PS (2003) Reduced order modeling of a two-dimensional flow with moving shocks. Comput Fluids 32(7):917–938
31. 31.
Nguyen NC, Peraire J (2008) An efficient reduced-order modeling approach for non-linear parametrized partial differential equations. Int J Numer Meth Eng 76(1):27–55
32. 32.
Noack BR, Morzynski M, Tadmor G (2011) Reduced-Order modelling for flow control. Springer, Berlin
33. 33.
Rabczuk T, Bordas SPA, Kerfriden P, Goury O (2012). A partitioned model order reduction approach to rationalise computational expenses in multiscale fracture mechanicsGoogle Scholar
34. 34.
Rozza G, Lassila T, Manzoni A (2011) Reduced basis approximation for shape optimization in thermal flows with a parametrized polynomial geometric map. In: Hesthaven JS, Ranquist EM (eds) Spectral and high order methods for partial differential equations., vol 76Springer, Berlin, pp 307–315
35. 35.
Ryckelynck D (2005) A priori hyperreduction method: an adaptive approach. J Comput Phys 202(1):346–366
36. 36.
Ryckelynck D (2009) Hyper-reduction of mechanical models involving internal variables. Int J Numer Meth Eng 77(1):75–89
37. 37.
Verhoeven A, Voss T, Astrid P, ter Maten EJW, Bechtold T (2007) Model order reduction for nonlinear problems in circuit simulation. PAMM 7(1):1021603–1021604
38. 38.
Verhoeven A, Maten J, Striebel M, Mattheij R (2009) Model order reduction for nonlinear ic models. In: Korytowski A, Malanowski K, Mitkowski W, Szymkat M (eds) System modeling and optimization, vol 312, IFIP advances in information and communication technology, Springer, Berlin, pp 476–491Google Scholar
39. 39.
Veroy K, Patera AT (2005). Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Int J Numer Meth Fluids, 47(8–9):773–788Google Scholar
40. 40.
Wang Z, Akhtar I, Borggaard J, Iliescu T (2011). Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison. arXiv:1106.3585
41. 41.
Wicke M, Stanton M, Treuille A. Modular bases for fluid dynamics. ACM Trans Graph, 28(3):39:1–39:8Google Scholar

© Springer International Publishing Switzerland 2014

## Authors and Affiliations

• Joan Baiges
• 1
• 2
• Ramon Codina
• 1
• 2
• Sergio R. Idelsohn
• 1
• 3
1. 1.Centre Internacional de Mètodes Numèrics a l’Enginyeria (CIMNE)BarcelonaSpain
2. 2.Universitat Politècnica de CatalunyaBarcelonaSpain
3. 3.Instituciò Catalana de Recerca i Estudis Avançats (ICREA)BarcelonaSpain