Numerical Analysis of a Method for Solving 2D Linear Isotropic Elastodynamics with Free Boundary Condition using Potentials and Finite Elements

Jorge Albella Martínez (1) , Sébastien Imperiale (2, 3) , Patrick Joly (4) , Jerónimo Rodríguez (1)

1 Departamento de Matemática Aplicada
2 M3DISIM - Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine
3 IP Paris - Institut Polytechnique de Paris
4 POEMS - Propagation des Ondes : Étude Mathématique et Simulation

When solving 2D linear elastodynamic equations in a 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 In [3, 2]. However in [4] the case of free surface boundary condition was proven to be unstable if a straight- forward 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 in [4] 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.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Jorge Albella Martínez, Sébastien Imperiale, Patrick Joly, Jerónimo Rodríguez. Numerical Analysis of a Method for Solving 2D Linear Isotropic Elastodynamics with Free Boundary Condition using Potentials and Finite Elements. Mathematics of Computation, inPress, ⟨10.1007/s10915-018-0768-9⟩. ⟨hal-02345808⟩