Solving 2D linear isotropic elastodynamics by means of scalar potentials: a new challenge for finite elements

Jorge Albella Martínez 1 Sébastien Imperiale 2, 3 Patrick Joly 4 Jerónimo Rodríguez 1
2 M3DISIM - Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine
LMS - Laboratoire de mécanique des solides, Inria Saclay - Ile de France
4 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
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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Jorge Albella Martínez, Sébastien Imperiale, Patrick Joly, Jerónimo Rodríguez. Solving 2D linear isotropic elastodynamics by means of scalar potentials: a new challenge for finite elements. Journal of Scientific Computing, Springer Verlag, 2018, ⟨10.1007/s10915-018-0768-9⟩. ⟨hal-01803536⟩

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