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Isogeometric Analysis: Mathematical and Implementational Aspects, with Applications

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Splines and PDEs: From Approximation Theory to Numerical Linear Algebra

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2219))

Abstract

Isogeometric analysis (IGA) is a recent and successful extension of classical finite element analysis. IGA adopts smooth splines, NURBS and generalizations to approximate problem unknowns, in order to simplify the interaction with computer aided geometric design (CAGD). The same functions are used to parametrize the geometry of interest. Important features emerge from the use of smooth approximations of the unknown fields. When a careful implementation is adopted, which exploit its full potential, IGA is a powerful and efficient high-order discretization method for the numerical solution of PDEs. We present an overview of the mathematical properties of IGA, discuss computationally efficient isogeometric algorithms, and present some significant applications.

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Notes

  1. 1.

    IGA of Cahn-Hilliard 4th-order model of phase separation is studied in [62, 63]; Kirchhoff-Love 4th-order model of plates and shells in [21, 26, 78, 79]; IGA of crack propagation is studied in [126], with 4th- and 6th-order gradient-enhanced theories of damage [127]; 4th-order phase-field fracture models are considered in [30] and [29], where higher-order convergence rates to sharp-interface limit solutions are numerically demonstrated.

  2. 2.

    The effect of regularity on the spectral behavior of isogeometric discretizations has been studied in [49, 71, 73].

  3. 3.

    It is possible to further reduce the memory requirements at the cost of increasing the number of computations. Indeed, note that it is not necessary to store the whole \(\mathscr {D} \), \( {\widetilde {v}} \) and \(\widetilde {\widetilde {v}} \) since w in Algorithm 4 can be computed component by component with on-the-fly calculation of the portion of \(\mathscr {D} \), \( {\widetilde {v}} \) and \(\widetilde {\widetilde {v}} \) that is needed).

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Acknowledgements

This chapter is a review of the mathematical results available on IGA. The presentation follows in particular [25, 48, 50, 57, 71, 73, 96]. We thank our colleagues and coauthors: Yuri Bazilevs, Lourenco Beirão da Veiga, Annalisa Buffa, Francesco Calabrò, Annabelle Collin, Austin Cottrell, John Evans, Alessandro Reali, Thomas Takacs, and Rafael Vázquez. Thomas J.R. Hughes was partially supported by the Office of Naval Research (Grant Nos. N00014-17-1-2119, N00014-17-1-2039, and N00014-13-1-0500), and by the Army Research Office (Grant No. W911NF-13-1-0220). Giancarlo Sangalli and Mattia Tani were partially supported by the European Research Council (ERC Consolidator Grant n.616563 HIGEOM). This support is gratefully acknowledged.

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Authors and Affiliations

  1. Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, USA

    Thomas J. R. Hughes

  2. Dipartimento di Matematica, Università di Pavia and Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” (CNR), Pavia, Italy

    Giancarlo Sangalli & Mattia Tani

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  1. Thomas J. R. Hughes
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Correspondence to Giancarlo Sangalli .

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

    Tom Lyche

  2. Department of Mathematics, University of Rome Tor Vergata, ROMA, Roma, Italy

    Carla Manni

  3. Department of Mathematics, University of Rome Tor Vergata, ROMA, Roma, Italy

    Hendrik Speleers

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Hughes, T.J.R., Sangalli, G., Tani, M. (2018). Isogeometric Analysis: Mathematical and Implementational Aspects, with Applications. In: Lyche, T., Manni, C., Speleers, H. (eds) Splines and PDEs: From Approximation Theory to Numerical Linear Algebra. Lecture Notes in Mathematics(), vol 2219. Springer, Cham. https://doi.org/10.1007/978-3-319-94911-6_4

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