Numerical analysis of a method for solving 2D linear isotropic elastodynamics with traction free boundary condition using potentials and finite elements
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- by Jorge Albella Martínez, Sébastien Imperiale, Patrick Joly and Jerónimo Rodríguez;
- Math. Comp. 90 (2021), 1589-1636
- DOI: https://doi.org/10.1090/mcom/3613
- Published electronically: April 5, 2021
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Abstract:
When solving 2D linear elastodynamic equations in homogeneous isotropic media, a Helmholtz decomposition of the displacement field decouples the equations into two scalar wave equations that only interact at the boundary. It is then natural to look for numerical schemes that independently solve the scalar equations and couple the solutions at the boundary. The case of rigid boundary condition was treated by Burel [Ph.D. thesis, Université Paris Sud-Paris XI (2014)] and Burel et al. [Numer. Anal. Appl. 5 (2012), pp. 136–143]. However the case of traction free boundary condition was proven by Martinez et al. [J. Sci. Comput. 77 (2018), pp. 1832–1873] to be unstable if a straightforward approach is used. Then an adequate functional framework as well as a time domain mixed formulation to circumvent these issues was presented. In this work we first review the formulation presented by Martinez et al. [J. Sci. Comput. 77 (2018), pp. 1832–1873] and propose a subsequent discretised formulation. We provide the complete stability analysis of the corresponding numerical scheme. Numerical results that illustrate the theory are also shown.References
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Bibliographic Information
- Jorge Albella Martínez
- Affiliation: Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Spain
- ORCID: 0000-0002-1335-2190
- Email: jorge.albella@usc.es
- Sébastien Imperiale
- Affiliation: Inria, Université Paris-Saclay, France; and LMS, Ecole Polytechnique, CNRS, Université Paris-Saclay, France
- ORCID: 0000-0001-5698-9653
- Email: sebastien.imperiale@inria.fr
- Patrick Joly
- Affiliation: Inria, Université Paris-Saclay, France; and UMA, Ensta, CNRS, Université Paris-Saclay, France
- MR Author ID: 234723
- Email: patrick.joly@inria.fr
- Jerónimo Rodríguez
- Affiliation: Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Spain; IMAT, Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Spain; and ITMATI, Campus Sur, 15706 Santiago de Compostela, Spain
- ORCID: 0000-0002-7367-4394
- Email: jeronimo.rodriguez@usc.es
- Received by editor(s): August 8, 2020
- Received by editor(s) in revised form: October 2, 2020
- Published electronically: April 5, 2021
- Additional Notes: The first and fourth authors were supported in part by FEDER/Ministerio de Ciencia, Innovación y Universidades and Agencia Estatal de Investigación through grants MTM2013-43745-R and MTM2017-86459-R and by Xunta de Galicia through grant ED431C 2017/60.
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 1589-1636
- MSC (2020): Primary 65M12, 65N30, 65N55, 35L05
- DOI: https://doi.org/10.1090/mcom/3613
- MathSciNet review: 4273110