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Solving 2D Linear Isotropic Elastodynamics by Means of Scalar Potentials: A New Challenge for Finite Elements

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Abstract

In this work we present a method for the computation of numerical solutions of 2D homogeneous isotropic elastodynamics equations by solving scalar wave equations. These equations act on the potentials of a Helmholtz decomposition of the displacement field and are decoupled inside the propagation domain. We detail how these equations are coupled at the boundary depending on the nature of the boundary condition satisfied by the displacement field. After presenting the case of rigid boundary conditions, that presents no specific difficulty, we tackle the challenging case of free surface boundary conditions that presents severe stability issues if a straightforward approach is used. We introduce an adequate functional framework as well as a time domain mixed formulation to circumvent these issues. Numerical results confirm the stability of the proposed approach.

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Acknowledgements

The research of the first and fourth authors was partially funded by FEDER and the Spanish Ministry of Science and Innovation through Grants MTM2013-43745-R and MTM2017-86459-R and by Xunta de Galicia through grant ED431C 2017/60.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, 15706, Santiago de Compostela, Spain

    Jorge Albella Martínez & Jerónimo Rodríguez

  2. Inria, Université Paris-Saclay, Paris, France

    Sébastien Imperiale & Patrick Joly

  3. LMS, Ecole Polytechnique, CNRS, Université Paris-Saclay, Paris, France

    Sébastien Imperiale

  4. UMA, Ensta, CNRS, Université Paris-Saclay, Paris, France

    Patrick Joly

  5. IMAT, Universidade de Santiago de Compostela, 15706, Santiago de Compostela, Spain

    Jerónimo Rodríguez

  6. ITMATI, Campus Sur, 15706, Santiago de Compostela, Spain

    Jerónimo Rodríguez

Authors
  1. Jorge Albella Martínez
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  2. Sébastien Imperiale
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  3. Patrick Joly
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  4. Jerónimo Rodríguez
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Corresponding author

Correspondence to Patrick Joly.

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Albella Martínez, J., Imperiale, S., Joly, P. et al. Solving 2D Linear Isotropic Elastodynamics by Means of Scalar Potentials: A New Challenge for Finite Elements. J Sci Comput 77, 1832–1873 (2018). https://doi.org/10.1007/s10915-018-0768-9

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  • DOI: https://doi.org/10.1007/s10915-018-0768-9

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