Journal of Scientific Computing

, Volume 77, Issue 3, pp 1832–1873 | Cite as

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

  • Jorge Albella Martínez
  • Sébastien Imperiale
  • Patrick JolyEmail author
  • Jerónimo Rodríguez


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.


Elastic wave propagation Helmholtz decomposition Potentials Stability of the evolution problem 



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. 1.Departamento de Matemática AplicadaUniversidade de Santiago de CompostelaSantiago de CompostelaSpain
  2. 2.Inria, Université Paris-SaclayParisFrance
  3. 3.LMS, Ecole Polytechnique, CNRS, Université Paris-SaclayParisFrance
  4. 4.UMA, Ensta, CNRS, Université Paris-SaclayParisFrance
  5. 5.IMAT, Universidade de Santiago de CompostelaSantiago de CompostelaSpain
  6. 6.ITMATI, Campus SurSantiago de CompostelaSpain

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