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Matrix Generation in Isogeometric Analysis by Low Rank Tensor Approximation

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Curves and Surfaces (Curves and Surfaces 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9213))

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Abstract

It has been observed that the task of matrix assembly in Isogeometric Analysis (IGA) is more challenging than in the case of traditional finite element methods. The additional difficulties associated with IGA are caused by the increased degree and the larger supports of the functions that occur in the integrals defining the matrix elements. Recently we introduced an interpolation-based approach that approximately transforms the integrands into piecewise polynomials and uses look-up tables to evaluate their integrals [17, 18]. The present paper relies on this earlier work and proposes to use tensor methods to accelerate the assembly process further. More precisely, we show how to represent the matrices that occur in IGA as sums of a small number of Kronecker products of auxiliary matrices that are defined by univariate integrals. This representation, which is based on a low-rank tensor approximation of certain parts of the integrands, makes it possible to achieve a significant speedup of the assembly process without compromising the overall accuracy of the simulation.

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Notes

  1. 1.

    Geometry + Simulation Modules, see www.gs.jku.at/gismo, also [10].

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Acknowledgement

This research was supported by the National Research Network “Geometry + Simulation” (NFN S117), funded by the Austrian Science Fund (FWF).

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Authors and Affiliations

  1. RICAM, Austrian Academy of Sciences, Linz, Austria

    Angelos Mantzaflaris, Bert Jüttler & Ulrich Langer

  2. Johannes Kepler University, Linz, Austria

    Bert Jüttler & Ulrich Langer

  3. MPI for Mathematics in the Sciences, Leipzig, Germany

    B. N. Khoromskij

Authors
  1. Angelos Mantzaflaris
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  2. Bert Jüttler
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  3. B. N. Khoromskij
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  4. Ulrich Langer
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Corresponding author

Correspondence to Bert Jüttler .

Editor information

Editors and Affiliations

  1. INRIA, Sophia Antipolis, Sophia Antipolis, France

    Jean-Daniel Boissonnat

  2. École Normale Supérieure, Paris, France

    Albert Cohen

  3. Arts Et Metiers ParisTech Lab de Sciences de l'Information, Paris, France

    Olivier Gibaru

  4. INSA de Rouen, LMI, Saint-Étienne-du-Rouvray, France

    Christian Gout

  5. Department of Mathematics, University of Oslo, Oslo, Norway

    Tom Lyche

  6. Université Joseph Fourier, Grenoble, France

    Marie-Laurence Mazure

  7. Department of Mathematics, Vanderbilt University, Nashville, Tennessee, USA

    Larry L. Schumaker

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Mantzaflaris, A., Jüttler, B., Khoromskij, B.N., Langer, U. (2015). Matrix Generation in Isogeometric Analysis by Low Rank Tensor Approximation. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2014. Lecture Notes in Computer Science(), vol 9213. Springer, Cham. https://doi.org/10.1007/978-3-319-22804-4_24

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  • DOI: https://doi.org/10.1007/978-3-319-22804-4_24

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22803-7

  • Online ISBN: 978-3-319-22804-4

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