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.
Mots clés : reduced basis method, reduced basis element method, domain decomposition, Schur complement, elliptic partial differential equations, a posteriori error estimation, component mode synthesis, parametrized systems
@article{M2AN_2013__47_1_213_0,
author = {Phuong Huynh, Dinh Bao and Knezevic, David J. and Patera, Anthony T.},
title = {A {Static} condensation {Reduced} {Basis} {Element} method : approximation and \protect\emph{a posteriori }error estimation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {213--251},
publisher = {EDP-Sciences},
volume = {47},
number = {1},
year = {2013},
doi = {10.1051/m2an/2012022},
zbl = {1276.65082},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2012022/}
}
TY - JOUR AU - Phuong Huynh, Dinh Bao AU - Knezevic, David J. AU - Patera, Anthony T. TI - A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 213 EP - 251 VL - 47 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012022/ DO - 10.1051/m2an/2012022 LA - en ID - M2AN_2013__47_1_213_0 ER -
%0 Journal Article %A Phuong Huynh, Dinh Bao %A Knezevic, David J. %A Patera, Anthony T. %T A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 213-251 %V 47 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012022/ %R 10.1051/m2an/2012022 %G en %F M2AN_2013__47_1_213_0
Phuong Huynh, Dinh Bao; Knezevic, David J.; Patera, Anthony T. A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 1, pp. 213-251. doi : 10.1051/m2an/2012022. http://www.numdam.org/articles/10.1051/m2an/2012022/
[1] , , and , 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. | MR | Zbl
[2] , and , 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. | MR | Zbl
[3] and . An automated multilevel substructuring method for eigenspace computation in linear elastodynamics. SIAM J. Sci. Comput. 25 (2004) 2084-2106. | MR | Zbl
[4] and , Galerkin lumped parameter methods for transient problems. Int. J. Numer. Methods Eng. 87 (2011) 943-961, doi: 10.1002/nme.3140. | MR | Zbl
[5] , , , , and , 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). | MR | Zbl
[6] , Component mode synthesis and eigenvalues of second order operators : discretization and algorithm. ESAIM : M2AN 26 (1992) 385-423. | Numdam | MR | Zbl
[7] , The condition number of the Schur complement in domain decompostion. Numer. Math. 83 (1999) 187-203. | MR | Zbl
[8] , , , and , A priori convergence of the greedy algorithm for the parametrized reduced basis. To appear in ESAIM : M2AN (2010). | Numdam | Zbl
[9] , and , 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). | MR | Zbl
[10] and , Coupling of substructures for dynamic analyses. AIAA J. 6 (1968) 1313-1319. | Zbl
[11] , , , and , Adaptive port reduction in static condensation, in MATHMOD 2012 - 7th Vienna International Conference on Mathematical Modelling (2012) (Submitted).
[12] , and , A reduced basis method for multiple electromagnetic scattering in three dimensions. Technical Report 2011-9, Scientific Computing Group, Brown University, Providence, RI, USA (2011).
[13] and , Matrix Computations. Johns Hopkins University Press (1996). | MR | Zbl
[14] and , Model reduction methods for dynamic analyses of large structures. Comput. Struct. 47 (1993) 735-749. | MR | Zbl
[15] and , A special finite element method based on component mode synthesis. ESAIM : M2AN 44 (2010) 401-420. | Numdam | MR | Zbl
[16] and , A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. | MR | Zbl
[17] , On the dynamic analysis of structural systems using component modes, in First AIAA Annual Meeting. Washington, DC, AIAA paper, No. 64-487 (1964).
[18] , , and , A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math. 345 (2007) 473-478. | MR | Zbl
[19] , 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. | MR | Zbl
[20] , and , Adaptive component mode synthesis in linear elasticity. Int. J. Numer. Methods Eng. 86 (2011) 829-844. | MR | Zbl
[21] , and , A new local reduced basis discontinuous galerkin approach for heterogeneous multiscale problems. C. R. Math. 349 (2011) 1233-1238. | MR | Zbl
[22] B.S. Kirk, J.W. Peterson, R.H. Stogner and G.F. Carey, libMesh : A C++ library for Parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 22 (2006) 237-254.
[23] and , A high-performance parallel implementation of the certified reduced basis method. Comput. Methods Appl. Mech. Eng. 200 (2011) 1455-1466. | Zbl
[24] Y Maday and EM Rønquist, The reduced basis element method : Application to a thermal fin problem. SIAM J. Sci. Comput. 26 (2004) 240-258. | MR | Zbl
[25] , and , A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. J. Sci. Comput. 17 (2002) 437-446. | MR | Zbl
[26] , A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales. J. Comput. Phys. 227 (2007) 9807-9822. | MR | Zbl
[27] , , , , and , Reliable real-time solution of parametrized partial differential equations : Reduced-basis output bounds methods. J. Fluids Eng. 124 (2002) 70-80.
[28] , and , Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15 (2008) 229-275. | MR
Cité par Sources :
