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A special finite element method based on component mode synthesis

Published online by Cambridge University Press:  04 February 2010

Ulrich L. Hetmaniuk  and
Richard B. Lehoucq
Ulrich L. Hetmaniuk
Affiliation:
Department of Applied Maths, University of Washington, Box 352420, Seattle, WA 98195-2420, USA. hetmaniu@u.washington.edu
Richard B. Lehoucq
Affiliation:
Sandia National Laboratories, P.O. Box 5800, MS 1320, Albuquerque, NM 87185-1320, USA. rblehou@sandia.gov
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Abstract

The goal of our paper is to introduce basis functions for the finite element discretization of a second order linear elliptic operator with rough or highly oscillating coefficients. The proposed basis functions are inspired by the classic idea of component mode synthesis and exploit an orthogonal decomposition of the trial subspace to minimize the energy. Numerical experiments illustrate the effectiveness of the proposed basis functions.

Keywords

Eigenvaluesmodal analysismultilevelsubstructuringdomain decompositiondimensional reductionfinite elements
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 3 , May 2010 , pp. 401 - 420
Copyright
© EDP Sciences, SMAI, 2010

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References

Babuška, I. and Osborn, J.E., Generalized finite element methods: Their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20 (1983) 510–536. CrossRef
Babuška, I., Caloz, G. and Osborn, J., Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31 (1994) 945–981. CrossRef
Babuška, I., Banerjee, U. and Osborn, J., On principles for the selection of shape functions for the generalized finite element method. Comput. Methods Appl. Mech. Engrg. 191 (2002) 5595–5629. CrossRef
Babuška, I., Banerjee, U. and Osborn, J.E., Generalized finite element methods – main ideas, results and perspective. Int. J. Comp. Meths. 1 (2004) 67–103. CrossRef
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. CrossRef
Bourquin, F., Component mode synthesis and eigenvalues of second order operators: Discretization and algorithm. ESAIM: M2AN 26 (1992) 385–423. CrossRef
Brezzi, F. and Marini, L., Augmented spaces, two-level methods, and stabilizing subgrids. Int. J. Numer. Meth. Fluids 40 (2002) 31–46. CrossRef
Craig, R.R., Jr. and M.C.C. Bampton, Coupling of substructures for dynamic analysis. AIAA J. 6 (1968) 1313–1319.
Y. Efendiev and T. Hou, Multiscale Finite Element Methods: Theory and Applications, Surveys and Tutorials in the Applied Mathematical Sciences 4. Springer, New York, USA (2009).
U. Hetmaniuk and R.B. Lehoucq, Multilevel methods for eigenspace computations in structural dynamics, in Domain Decomposition Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng. 55, Springer-Verlag (2007) 103–114.
Hou, T. and Wu, X., A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. CrossRef
Hurty, W.C., Vibrations of structural systems by component-mode synthesis. J. Eng. Mech. Division ASCE 86 (1960) 51–69.
Nolen, J., Papanicolaou, G. and Pironneau, O., A framework for adaptive multiscale methods for elliptic problems. Multiscale Model. Simul. 7 (2008) 171–196. CrossRef
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations – Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, UK (1999).