| Issue |
ESAIM: M2AN
Volume 51, Number 1, January-February 2017
|
|
|---|---|---|
| Page(s) | 147 - 163 | |
| DOI | https://doi.org/10.1051/m2an/2016019 | |
| Published online | 28 November 2016 | |
Generalized finite element methods for quadratic eigenvalue problems∗,∗∗
1 Department of Mathematics, Chalmers University of Technology
and University of Gothenburg, Sweden.
axel@chalmers.se
2 Insitute for Numerical Simulation, Rheinische
Friedrich-Wilhelms-Universität, Bonn, Germany.
Received:
19
October
2015
Revised:
5
February
2016
Accepted:
8
March
2016
We consider a large-scale quadratic eigenvalue problem (QEP), formulated using P1 finite elements on a fine scale reference mesh. This model describes damped vibrations in a structural mechanical system. In particular we focus on problems with rapid material data variation, e.g., composite materials. We construct a low dimensional generalized finite element (GFE) space based on the localized orthogonal decomposition (LOD) technique. The construction involves the (parallel) solution of independent localized linear Poisson-type problems. The GFE space is used to compress the large-scale algebraic QEP to a much smaller one with a similar modeling accuracy. The small scale QEP can then be solved by standard techniques at a significantly reduced computational cost. We prove convergence with rate for the proposed method and numerical experiments confirm our theoretical findings.
Mathematics Subject Classification: 65N30 / 65N25 / 65N15
Key words: Quadratic eigenvalue problem / finite element / localized orthogonal decomposition
A. Målqvist is supported by the Swedish Research Council and the Swedish Foundation for Strategic Research.
D. Peterseim is supported by the Hausdorff Center for Mathematics Bonn and by Deutsche Forschungsgemeinschaft in the Priority Program 1748 “Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis” under the project “Adaptive isogeometric modeling of propagating strong discontinuities in heterogeneous materials”.
© EDP Sciences, SMAI 2016
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