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An efficient computational framework for reducedbasis approximation and a posteriori error estimation of parametrized Navier–Stokes flows

Published online by Cambridge University Press:  15 July 2014

Andrea Manzoni
Andrea Manzoni*
Affiliation:
SISSA Mathlab – International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy. amanzoni@sissa.it
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Abstract

We present the current Reduced Basis framework for the efficient numerical approximation of parametrized steady Navier–Stokes equations. We have extended the existing setting developed in the last decade (see e.g. [S. Deparis, SIAM J. Numer. Anal. 46 (2008) 2039–2067; A. Quarteroni and G. Rozza, Numer. Methods Partial Differ. Equ. 23 (2007) 923–948; K. Veroy and A.T. Patera, Int. J. Numer. Methods Fluids 47 (2005) 773–788]) to more general affine and nonaffine parametrizations (such as volume-based techniques), to a simultaneous velocity-pressure error estimates and to a fully decoupled Offline/Online procedure in order to speedup the solution of the reduced-order problem. This is particularly suitable for real-time and many-query contexts, which are both part of our final goal. Furthermore, we present an efficient numerical implementation for treating nonlinear advection terms in a convenient way. A residual-based a posteriori error estimation with respect to a truth, full-order Finite Element approximation is provided for joint pressure/velocity errors, according to the Brezzi–Rappaz–Raviart stability theory. To do this, we take advantage of an extension of the Successive Constraint Method for the estimation of stability factors and of a suitable fixed-point algorithm for the approximation of Sobolev embedding constants. Finally, we present some numerical test cases, in order to show both the approximation properties and the computational efficiency of the derived framework.

Keywords

Reduced Basis Methodparametrized Navier–Stokes equationssteady incompressible fluidsa posteriori error estimationapproximation stability
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014 , pp. 1199 - 1226
Copyright
© EDP Sciences, SMAI, 2014

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References

Barrault, M., Maday, Y., Nguyen, N.C. and Patera, A.T., An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris 339 (2004) 667–672. Google Scholar
Biswas, G., Breuer, M. and Durst, F., Backward-facing step flows for various expansion ratios at low and moderate Reynolds numbers. J. Fluids Eng. 126 (2004) 362–374. Google Scholar
Brezzi, F., On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO. Anal. Numér. 2 (1974) 129–151. Google Scholar
Brezzi, F., Rappaz, J. and Raviart, P.A., Finite dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions. Numer. Math. 36 (1980) 1–25. Google Scholar
G. Caloz and J. Rappaz, Numerical analysis for nonlinear and bifurcation problems. In vol. 5, Techniques of Scientific Computing (Part 2). Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions. Elsevier Science B.V. (1997) 487–637.
Canuto, C., Tonn, T. and Urban, K., A posteriori error analysis of the reduced basis method for non-affine parameterized nonlinear pdes. SIAM J. Numer. Anal. 47 (2009) 2001–2022. Google Scholar
Deparis, S., Reduced basis error bound computation of parameter-dependent Navier–Stokes equations by the natural norm approach. SIAM J. Numer. Anal. 46 (2008) 2039–2067. Google Scholar
Deparis, S. and Rozza, G., Reduced basis method for multi-parameter-dependent steady Navier-Stokes equations: Applications to natural convection in a cavity. J. Comput. Phys. 228 (2009) 4359–437. CrossRefGoogle Scholar
H.C. Elman, D.J. Silvester and A.J. Wathen, Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics. Series in Numer. Math. Sci. Comput. Oxford Science Publications, Clarendon Press, Oxford (2005).
Gerner, A.-L. and Veroy, K., Reduced basis a posteriori error bounds for the Stokes equations in parametrized domains: a penalty approach. Math. Models Methods Appl. Sci. 21 (2010) 2103–2134. Google Scholar
P.M. Gresho and R.L. Sani, Incompressible Flow and the Finite Element Method: Advection-Diffusion and Isothermal Laminar Flow.John Wiley & Sons (1998).
Herrero, H., Maday, Y. and Pla, F., RB (reduced basis) for RB (Rayleigh-Bénard). Comput. Methods Appl. Mech. Engrg. 261–262 (2013) 132–141. CrossRefGoogle Scholar
Huynh, D.B.P., Knezevic, D.J., Chen, Y., Hesthaven, J.S. and Patera, A.T., A natural-norm successive constraint method for inf-sup lower bounds. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1963–1975. Google Scholar
Ito, K. and Ravindran, S.S., A reduced order method for simulation and control of fluid flows. J. Comput. Phys. 143 (1998) 403–425. Google Scholar
Lassila, T., Manzoni, A., Quarteroni, A. and Rozza, G., Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty. ESAIM: M2AN 47 (2013) 1107–1131. Google Scholar
T. Lassila, A. Manzoni, A. Quarteroni and G. Rozza, Model order reduction in fluid dynamics: challenges and perspectives. In vol. 9, Reduced Order Methods for Modeling and Computational Reduction. Edited by A. Quarteroni and G. Rozza. Springer MS&A Series (2014) 235–274.
A. Manzoni, Reduced models for optimal control, shape optimization and inverse problems in haemodynamics. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2012).
A. Manzoni and F. Negri, Rigorous and heuristic strategies for the approximation of stability factors in nonlinear parametrized PDEs. Technical report MATHICSE 8.2014: http://mathicse.epfl.ch/, submitted (2014).
Manzoni, A., Quarteroni, A. and Rozza, G., Model reduction techniques for fast blood flow simulation in parametrized geometries. Int. J. Numer. Methods Biomed. Engrg. 28 (2012) 604–625. Google Scholar
Manzoni, A., Quarteroni, A. and Rozza, G., Shape optimization of cardiovascular geometries by reduced basis methods and free-form deformation techniques. Int. J. Numer. Methods Fluids 70 (2012) 646–670. Google Scholar
N.C. Nguyen, K. Veroy and A.T. Patera, Certified real-time solution of parametrized partial differential equations. Handbook of Materials Modeling. Edited by S. Yip. Springer, The Netherlands (2005) 1523–1558.
Peterson, J.S., The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Statis. Comput. 10 (1989) 777–786. Google Scholar
Quarteroni, A. and Rozza, G., Numerical solution of parametrized Navier-Stokes equations by reduced basis methods. Numer. Methods Partial Differ. Equ. 23 (2007) 923–948. Google Scholar
A. Quarteroni, G. Rozza and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations in industrial applications. J. Math. Ind. 1 (2011).
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations 1st edition. Springer-Verlag, Berlin-Heidelberg (1994).
Rozza, G., Huynh, D.B.P. and Manzoni, A., Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants. Numer. Math. 125 (2013) 115–152. Google Scholar
Rozza, G., Huynh, D.B.P. and Patera, A.T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Engrg. 15 (2008) 229–275. Google Scholar
Rozza, G. and Veroy, K., On the stability of reduced basis methods for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Engrg. 196 (2007) 1244–1260. Google Scholar
Sen, S., Veroy, K., Huynh, D.B.P., Deparis, S., Nguyen, N.C. and Patera, A.T., “Natural norm” a posteriori error estimators for reduced basis approximations. J. Comput. Phys. 217 (2006) 37–62. Google Scholar
R. Temam, Navier-Stokes Equations. AMS Chelsea, Providence, Rhode Island (2001).
Veroy, K. and Patera, A.T., 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 (2005) 773–788. CrossRefGoogle Scholar
Yano, M. and Patera, A.T., A space-time variational approach to hydrodynamic stability theory. Proc. R. Soc. A 469 (2013) 0036. CrossRefGoogle Scholar