Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T11:54:04.944Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduced Basis Approximation and Error Bounds for Potential Flows in Parametrized Geometries

Published online by Cambridge University Press:  20 August 2015

Gianluigi Rozza
Gianluigi Rozza*
Affiliation:
Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-264, 77 Massachusetts Avenue, Cambridge MA, 02142-4307, USA Modelling and Scientific Computing, Ecole Polytechnique Fédérale de Lausanne, Station 8-MA, CH 1015, Lausanne, Switzerland
*
*Corresponding author.Email:[email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for potential flows in affinely parametrized geometries. We review the essential ingredients: i) a Galerkin projection onto a low-dimensional space associated with a smooth “parametric manifold” in order to get a dimension reduction; ii) an efficient and effective greedy sampling method for identification of optimal and numerically stable approximations to have a rapid convergence; iii) an a posteriori error estimation procedure: rigorous and sharp bounds for the linear-functional outputs of interest and over the potential solution or related quantities of interest like velocity and / or pressure; iv) an Offline-Online computational decomposition strategies to achieve a minimum marginal computational cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present three illustrative results for inviscid potential flows in parametrized geometries representing a Venturi channel, a circular bend and an added mass problem.

Keywords

65Y2076B9935Q3565N15Reduced basis approximationerror boundspotential flowsGalerkin methoda posteriori error estimationparametrized geometries
Type
Review Article
Information
Communications in Computational Physics , Volume 9 , Issue 1 , January 2011 , pp. 1 - 48
Copyright
Copyright © Global Science Press Limited 2011

References

[1]Almroth, B.O., Stern, P., and Brogan, F.A.. Automatic choice of global shape functions in structural analysis. AIAA Journal, 16:525–528, 1978.Google Scholar
[2]Reduced Basis at MIT. http://augustine.mit.edu/methodology.htm. MIT, Cambridge, 2007–2010. Massachusetts Institute of Technology.Google Scholar
[3]Balmes, E.. Parametric falies of reduced finite element models. theory and applications. Mechanical Systems and Signal Processing, 44(170):283–301, 1996.Google Scholar
[4]Barrett, A. and Reddien, G.. On the reduced basis method. Zeitschrift für Angewandte Mathematik und Mechanik, 75(7):543–549, 1995.CrossRefGoogle Scholar
[5]Bertin, J.J.. Aerodynamics for Engineers. Prentice Hall, 2002.Google Scholar
[6]Bui-Thanh, T., Willcox, K., and Ghattas, O.. Model reduction for large scale systems with high-dimensional parametric input space. In Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, volume AIAA Paper 2007-2049, 2007.Google Scholar
[7]Elman, H.C., Silvester, D.J., and Wathen, A.J.. Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Oxford university Press, 2005.CrossRefGoogle Scholar
[8]Fink, J.P. and Rheinboldt, W.C.. On the error behavior of the reduced basis technique for nonlinear finite element approximations. Zeitschrift für Angewandte Mathematik und Mechanik, 63:21–28, 1983.Google Scholar
[9]Fink, J.P. and Rheinboldt, W.C.. Local error estimates for parametrized non-linear equations. SIAM Journal on Numerical Analysis, 22:729–735, 1985.Google Scholar
[10]Grepl, M.A., Maday, Y., Nguyen, N.C., and Patera, A.T.. Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. Mathematical Modelling and Numerical Analysis, 41(3):575–605, 2007.Google Scholar
[11]Haasdonk, B., Ohlberger, M., and Rozza, G.. A reduced basis method for evolution schemes with parameter-dependent explicit operators. Electronic Transactions on Numerical Analysis, 32:145–161, 2008.Google Scholar
[12]Huynh, D.B.P., Nguyen, N.C., Rozza, G., and Patera, A.T.. Documentation for rbMIT Software: I. Reduced Basis (RB) for Dummies. Massachusetts Institute of Technology, 2007–2010. Available at http://augustine.mit.edu together with the rbMIT Software package.Google Scholar
[13]Huynh, D.B.P., Rozza, G., Sen, S., and Patera, A.T.. A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. Comptes Rendus de l’Académie des Sciences Paris, Sér I 345:473–478, 2007.Google Scholar
[14]Ito, K. and Ravindran, S.S.. A reduced basis method for control problems governed by PDEs. International Series of Numerical Mathematics, 126:153–168, 1998.Google Scholar
[15]Ito, K. and Ravindran, S.S.. A reduced-order method for simulation and control of fluid flow. Journal of Computational Physics, 143(2):403–425, 1998.Google Scholar
[16]Ito, K. and Ravindran, S.S.. Reduced basis method for optimal control of unsteady viscous flows. International Journal of Computational Fluid Dynamics, 15:97–113, 2001.Google Scholar
[17]Ito, K. and Schroeter, J.D.. Reduced order feedback synthesis for viscous incompressible flows. Mathematical And Computer Modelling, 33(1-3):173–192, 2001.Google Scholar
[18]Lin Lee, M.Y.. Estimation of the error in the reduced-basis method solution of differential algebraic equations. SIAM Journal on Numerical Analysis, 28:512–528, 1991.Google Scholar
[19]Løvgren, A.E., Maday, Y., and Rønquist, E.M.. A reduced basis element method for the steady Stokes problem. Mathematical Modelling and Numerical Analysis, 40(3):529–552, 2006.Google Scholar
[20]Machiels, L., Maday, Y., Oliveira, I.B., Patera, A.T., and Rovas, D.V.. Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. Comptes Rendus de l’Académie des Sciences Paris, Sér I 331(2):153–158, 2000.Google Scholar
[21]Maday, Y., Machiels, L., Patera, A.T., and Rovas, D.V.. Blackbox reduced-basis output bound methods for shape optimization. In Proceedings of the 12th international Domain Decomposition Conference, 2000.Google Scholar
[22]Maday, Y., Patera, A.T., and Rovas, D.V.. A blackbox reduced-basis output bound method for noncoercive linear problems. In Lions, J.LCioranescu, D., editor, Nonlinear partial differential equations and their applications, College de France Seminar, vol. XIV, pages 535–569. Elseviever, Amsterdam, 2002.Google Scholar
[23]Maday, Y., Patera, A.T., and Turnici, G.. Global a priori convergence theory for reduced basis approximation of single-parameter symmetric coercive elliptic partial differential equations. Comptes Rendus de l’Académie des Sciences Paris, 2002.Google Scholar
[24]Maday, Y., Patera, A.T., and Turnici, G.. A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. Journal of Scientific Computing, 2002.Google Scholar
[25]Newman, J.N.. Marine Hydrodynamics. MIT Press, 1977.Google Scholar
[26]Nguyen, N.C., Veroy, K., and Patera, A.T.. Certified real-time solution of parametrized partial differential equations. In Yip, S., editor, Handbook of Materials Modeling, pages 1523–1558. Springer, 2005.Google Scholar
[27]Noor, A.K.. Recent advances in reduction methods for nonlinear problems. Computers & Structures, 13:31–44, 1981.Google Scholar
[28]Noor, A.K.. On making large nonlinear problems small. Computer Methods in Applied Mechanics and Engineering, 34:955–985, 1982.Google Scholar
[29]Noor, A.K., Andresen, C.M., and Tanner, J.A.. Exploiting symmetries in the modeling and analysis of tires. Computer Methods in Applied Mechanics and Engineering, 63:37–81, 1987.Google Scholar
[30]Noor, A.K., Balch, C.D., and Shibut, M.A.. Reduction methods for non-linear steady-state thermal analysis. International Journal for Numerical Methods in Engineering, 20:1323–1348, 1984.Google Scholar
[31]Noor, A.K. and Peters, J.M.. Reduced basis technique for nonlinear analysis of structures. AIAA Journal, 18(4):455–462, 1980.Google Scholar
[32]Noor, A.K. and Peters, J.M.. Multiple-parameter reduced basis technique for bifurcation and post-buckling analysis of composite plates. International Journal for Numerical Methods in Engineering, 19:1783–1803, 1983.Google Scholar
[33]Noor, A.K. and Peters, J.M.. Recent advances in reduction methods for instability analysis of structures. Computers & Structures, 16:67–80, 1983.CrossRefGoogle Scholar
[34]Noor, A.K., Peters, J.M., and Andersen, C.M.. Mixed models and reduction techniques for large-rotation nonlinear problems. Computer Methods in Applied Mechanics and Engineering, 44:67–89, 1984.Google Scholar
[35]Oliveira, I.B. and Patera, A.T.. Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations. Optimization and Engineering, 8:43–65, 2007.Google Scholar
[36]Panton, R. L.. Incompressible Flow. John Wiley & Sons, Inc., 3rd edition, 2005.Google Scholar
[37]Patera, A.T. and Rozza, G.. Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. To appear in MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2007. Massachusetts Institute of Technology, Version 1.0.Google Scholar
[38]Peterson, J.S.. The reduced basis method for incompressible viscous flow calculations. SIAM Journal on Scientific and Statistical Computing, 10(4):777–786, 1989.Google Scholar
[39]Porsching, T.A.. Estimation of the error in the reduced basis method solution of nonlinear equations. Mathematics of Computation, 45(172):487–496, 1985.Google Scholar
[40]Porsching, T.A. and Lin Lee, M.Y.. The reduced-basis method for initial value problems. SIAM Journal on Numerical Analysis, 24:1277–1287, 1987.Google Scholar
[41]Prud’homme, C., Rovas, D.V., Veroy, K., Machiels, L., maday, Y., Patera, A.T., and Turinici, G.. Reliable real-time solution of parametrized partial differential equations. Journal of Fluids Engineering, 124:70–80, 2002.CrossRefGoogle Scholar
[42]Quarteroni, A., Rozza, G., and Quaini, A.. Reduced basis methods for optimal control of advection-diffusion problems. In Fitzgibbon, W., Hoppe, R., and Periaux, J., editors, Advances in Numerical Mathematics, pages 193–216. Moscow, Russia and Houston, USA, 2007.Google Scholar
[43]Quarteroni, A. and Valli, A.. Numerical Approximation of Partial Differential Equations. Springer-Verlag Berlin, 1997.Google Scholar
[44]rbMIT Library. http://augustine.mit.edu/methodology/methodologyjrbmit_system.htm. MIT, Cambridge, 2007-2010. Massachusetts Institute of Technology.Google Scholar
[45]Rheinboldt, W.C.. On the theory and error estimation of the reduced basis method for mulit-parameter problems. Nonlinear Analysis, Theory, Methods and Applications, 21(11):849–858, 1993.Google Scholar
[46]Rovas, D.V.. Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations. PhD thesis, Massachusetts Institute of Technology, 2003.Google Scholar
[47]Rozza, G.. Reduced basis methods for Stokes equations in domains with non-affine parameter dependence. Computing and Visualization in Science, 12(1):23–35, 2009.Google Scholar
[48]Rozza, G.. Shape Design by Optimal Flow Control and Reduced Basis Techniques: Applications to Bypass Configurations in Haemodynamics PhD thesis, Ecole Polytechnique Federale de Lausanne, November 2005.Google Scholar
[49]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: application to transport and continuum mechanics. Archives of Computational Methods in Engineering, 15(3):229–275, 2008.Google Scholar
[50]Rozza, G. and Veroy, K.. On the stability of the reduced basis method for Stokes equations in parametrized domains. Computer methods in applied mechanics and engineering, 196:1244–1260, 2007.Google Scholar
[51]Schröder, W.. Fluidmechanik - Aachener Beiträge zur Strömungsmechanik, Band 7. Wissenschaftsverlag Mainz in Aachen, 2004.Google Scholar
[52]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. Journal of Computational Physics, 217:37–62, 2006.Google Scholar
[53]Veroy, K. and Patera, A.T.. Certified real-time solutions of the parametrized steady incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 47:773–788, 2005.Google Scholar
[54]Veroy, K., Prud’homme, C., Rovas, D.V., and Patera, A.T.. A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In Proceedings of the 16th AIAA computational fluid dynamics conference, volume Paper 2003-3847, 2003.CrossRefGoogle Scholar