Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-19T20:33:31.511Z Has data issue: false hasContentIssue false
  • English
  • Français

A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation

Published online by Cambridge University Press:  23 November 2012

Dinh Bao Phuong Huynh ,
David J. Knezevic  and
Anthony T. Patera
Dinh Bao Phuong Huynh
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, 02139 MA, USA. huynh@mit.edu
David J. Knezevic
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, 02139 MA, USA. huynh@mit.edu School of Engineering and Applied Sciences, Harvard University, Cambridge, 02138 MA, USA; dknezevic@seas.harvard.edu; patera@mit.edu
Anthony T. Patera
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, 02139 MA, USA. huynh@mit.edu
Get access

Abstract

We propose a new reduced basis element-cum-component mode synthesis approach for parametrized elliptic coercive partial differential equations. In the Offline stage we construct a Library of interoperable parametrized reference components relevant to some family of problems; in the Online stage we instantiate and connect reference components (at ports) to rapidly form and query parametric systems. The method is based on static condensation at the interdomain level, a conforming eigenfunction “port” representation at the interface level, and finally Reduced Basis (RB) approximation of Finite Element (FE) bubble functions at the intradomain level. We show under suitable hypotheses that the RB Schur complement is close to the FE Schur complement: we can thus demonstrate the stability of the discrete equations; furthermore, we can develop inexpensive and rigorous (system-level) a posteriori error bounds. We present numerical results for model many-parameter heat transfer and elasticity problems with particular emphasis on the Online stage; we discuss flexibility, accuracy, computational performance, and also the effectivity of the a posteriori error bounds.

Keywords

Reduced Basis methodReduced Basis Element methoddomain decompositionSchur complementelliptic partial differential equationsa posteriori error estimationcomponent mode synthesisparametrized systems
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 213 - 251
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antil, H., Heinkenschloss, M., Hoppe, R.H.W. and Sorensen, D.C., Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables. Comput. Visualization Sci. 13 (2010) 249–264. Google Scholar
Antil, H., Heinkenschloss, M. and Hoppe, R.H.W., Domain decomposition and balanced truncation model reduction for shape optimization of the Stokes system. Optim. Methods Softw. 26 (2011) 643–669, doi: 10.1080/10556781003767904. Google Scholar
Bennighof, J.K. and Lehoucq, R.B.. An automated multilevel substructuring method for eigenspace computation in linear elastodynamics. SIAM J. Sci. Comput. 25 (2004) 2084–2106. Google Scholar
Bermúdez, A. and Pena, F., Galerkin lumped parameter methods for transient problems. Int. J. Numer. Methods Eng. 87 (2011) 943–961, doi: 10.1002/nme.3140. Google Scholar
P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods. Technical Report, Aachen Institute for Advanced Study in Computational Engineering Science, preprint : AICES-2010/05-2 (2010).
Bourquin, F., Component mode synthesis and eigenvalues of second order operators : discretization and algorithm. ESAIM : M2AN 26 (1992) 385–423. Google Scholar
Brenner, S.C., The condition number of the Schur complement in domain decompostion. Numer. Math. 83 (1999) 187–203. Google Scholar
A. Buffa, Y. Maday, A.T. Patera, C. Prud’homme and G. Turinici, A priori convergence of the greedy algorithm for the parametrized reduced basis. To appear in ESAIM : M2AN (2010).
Y. Chen, J.S. Hesthaven and Y. Maday, A Seamless Reduced Basis Element Methods for 2D Maxwell’s Problem : An Introduction, edited by J. Hesthaven and E.M. Rønquist, in Spectral and High Order Methods for Partial Differential Equations-Selected papers from the ICASOHOM’09 Conference 76 (2011).
Craig, R. and Bampton, M., Coupling of substructures for dynamic analyses. AIAA J. 6 (1968) 1313–1319. Google Scholar
J.L. Eftang, D.B.P. Huynh, D.J. Knezevic, E.M. Rønquist and A.T. Patera, Adaptive port reduction in static condensation, in MATHMOD 2012 – 7th Vienna International Conference on Mathematical Modelling (2012) (Submitted).
M. Ganesh, J.S. Hesthaven and B. Stamm, A reduced basis method for multiple electromagnetic scattering in three dimensions. Technical Report 2011-9, Scientific Computing Group, Brown University, Providence, RI, USA (2011).
G. Golub and C. van Loan, Matrix Computations. Johns Hopkins University Press (1996).
Haggblad, B. and Eriksson, L., Model reduction methods for dynamic analyses of large structures. Comput. Struct. 47 (1993) 735–749. Google Scholar
Hetmaniuk, U.L. and Lehoucq, R.B., A special finite element method based on component mode synthesis. ESAIM : M2AN 44 (2010) 401–420. Google Scholar
Hou, T.Y. and Wu, X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. Google Scholar
W.C. Hurty, On the dynamic analysis of structural systems using component modes, in First AIAA Annual Meeting. Washington, DC, AIAA paper, No. 64-487 (1964).
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. C. R. Math. 345 (2007) 473–478. Google Scholar
Iapichino, L., Quarteroni and G.A., Rozza, A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks. Comput. Methods Appl. Mech. Eng. 221-222 (2012) 63–82. Google Scholar
Jakobsson, H., Beingzon, F. and Larson, M.G., Adaptive component mode synthesis in linear elasticity. Int. J. Numer. Methods Eng. 86 (2011) 829–844. Google Scholar
Kaulmann, S., Ohlberger, M. and Haasdonk, B., A new local reduced basis discontinuous galerkin approach for heterogeneous multiscale problems. C. R. Math. 349 (2011) 1233–1238. Google Scholar
Kirk, B.S., Peterson, J.W., Stogner, R.H. and Carey, G.F., libMesh : A C++ library for Parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 22 (2006) 237–254. Google Scholar
Knezevic, D.J. and Peterson, J.W., A high-performance parallel implementation of the certified reduced basis method. Comput. Methods Appl. Mech. Eng. 200 (2011) 1455–1466. Google Scholar
Maday, Y and Rønquist, EM, The reduced basis element method : Application to a thermal fin problem. SIAM J. Sci. Comput. 26 (2004) 240–258. Google Scholar
Maday, Y., Patera, A.T. and Turinici, G., A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. J. Sci. Comput. 17 (2002) 437–446. Google Scholar
Nguyen, N.C., A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales. J. Comput. Phys. 227 (2007) 9807–9822. Google Scholar
Prud’homme, C., Rovas, D., Veroy, K., Maday, Y., Patera, A.T. and Turinici, G., Reliable real-time solution of parametrized partial differential equations : Reduced-basis output bounds methods. J. Fluids Eng. 124 (2002) 70–80. 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 Eng. 15 (2008) 229–275. Google Scholar