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Geometrically Unfitted Finite Element Methods and Applications pp 307–329Cite as

Well Conditioned Extended Finite Elements and Vector Level Sets for Three-Dimensional Crack Propagation

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

Abstract

A stable extended finite element method (XFEM) is combined to a three dimensional version of the vector level set method (Ventura et al., Int. J. Numer. Methods Eng. 58(10):1571–1592, 2003) to solve non-planar three-dimensional (3D) crack propagation problems.

The proposed XFEM variant is based on an extension of the degree of freedom gathering technique (Laborde et al., Int. J. Numer. Methods Eng. 64(3):354–381, 2005; Agathos et al., Int. J. Numer. Methods Eng. 105(9):643–677, 2016) which allows the use of geometrical enrichment in 3D without conditioning problems. The method is also combined to weight function blending and enrichment function shifting (Fries, Int. J. Numer. Methods Eng. 75(November 2007):503–532, 2008; Ventura et al. Int. J. Numer. Methods Eng. 77(July 2008):1–29, 2009) in order to remove blending errors and further improve conditioning. The improved conditioning results in a decrease in the number of iterations required to solve the resulting linear systems which for the cases studied ranges from 50% up to several orders of magnitude.

The propagating crack is represented using a 3D version of the level set method (Ventura et al., Int. J. Numer. Methods Eng. 58(10):1571–1592, 2003). In this method at any propagation step, the crack front is represented as an ordered series of line segments and the crack surface as a sequence of four sided bilinear surfaces. Level set functions are obtained by projecting points on those surfaces and line segments thus employing only geometrical operations and avoiding the solution of differential evolution equations.

The combination of the aforementioned methods is able to handle crack propagation problems providing improved accuracy, reduced computational cost and simplified implementation.

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Acknowledgements

Stéphane P.A. Bordas thanks the financial support of the European Research Council Starting Independent Research Grant (ERC Stg grant agreement No. 279578) entitled “Towards real time multiscale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery.”

Stéphane P.A. Bordas is also grateful for the support of the Fonds National de la Recherche Luxembourg FWO-FNR grant INTER/FWO/15/10318764.

Author information

Authors and Affiliations

  1. Research Unit in Engineering Science, Luxembourg University, 6 rue Richard Coudenhove-Kalergi, L-1359, Luxembourg, Luxembourg

    Konstantinos Agathos

  2. Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Giulio Ventura

  3. Institute of Structural Engineering, ETH Zürich, Stefano-Franscini-Platz 5, CH-8093, Zürich, Switzerland

    Eleni Chatzi

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

    Stéphane P. A. Bordas

  5. School of Engineering, Cardiff University, Cardiff Wales, UK

    Stéphane P. A. Bordas

Authors
  1. Konstantinos Agathos
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  2. Giulio Ventura
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  3. Eleni Chatzi
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  4. Stéphane P. A. Bordas
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Corresponding author

Correspondence to Konstantinos Agathos .

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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Agathos, K., Ventura, G., Chatzi, E., Bordas, S.P.A. (2017). Well Conditioned Extended Finite Elements and Vector Level Sets for Three-Dimensional Crack Propagation. 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_10

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