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Cut Finite Element Methods for Linear Elasticity Problems

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Geometrically Unfitted Finite Element Methods and Applications

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 121))

Abstract

We formulate a cut finite element method for linear elasticity based on higher order elements on a fixed background mesh. Key to the method is a stabilization term which provides control of the jumps in the derivatives of the finite element functions across faces in the vicinity of the boundary. We then develop the basic theoretical results including error estimates and estimates of the condition number of the mass and stiffness matrices. We apply the method to the standard displacement problem, the frequency response problem, and the eigenvalue problem. We present several numerical examples including studies of thin bending dominated structures relevant for engineering applications. Finally, we develop a cut finite element method for fibre reinforced materials where the fibres are modeled as a superposition of a truss and a Euler-Bernoulli beam. The beam model leads to a fourth order problem which we discretize using the restriction of the bulk finite element space to the fibre together with a continuous/discontinuous finite element formulation. Here the bulk material stabilizes the problem and it is not necessary to add additional stabilization terms.

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Acknowledgements

This research was supported in part by the Swedish Foundation for Strategic Research Grant No. AM13-0029, the Swedish Research Council Grant No. 2013-4708, and the Swedish strategic research programme eSSENCE.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Jönköping University, SE-551 11, Jönköping, Sweden

    Peter Hansbo

  2. Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87, Umeå, Sweden

    Mats G. Larson & Karl Larsson

Authors
  1. Peter Hansbo
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  2. Mats G. Larson
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  3. Karl Larsson
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Corresponding author

Correspondence to Mats G. Larson .

Editor information

Editors and Affiliations

  1. Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, Luxembourg

    Stéphane P. A. Bordas

  2. Department of Mathematics, University College London, United Kingdom

    Erik Burman

  3. Department of Mathematics and Mathematical Statistics, Umeå University, Sweden

    Mats G. Larson

  4. Department of Mathematics, University of Houston, USA

    Maxim A. Olshanskii

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Hansbo, P., Larson, M.G., Larsson, K. (2017). Cut Finite Element Methods for Linear Elasticity Problems. In: Bordas, S., Burman, E., Larson, M., Olshanskii, M. (eds) Geometrically Unfitted Finite Element Methods and Applications. Lecture Notes in Computational Science and Engineering, vol 121. Springer, Cham. https://doi.org/10.1007/978-3-319-71431-8_2

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