On the numerical solution of the elastodynamic problem by a boundary integral equation method

TL;DR

This paper addresses the Dirichlet initial boundary value problem for the time-dependent elastodynamic equation in an unbounded 2D exterior domain, $u_{tt}=\\Delta^*u$ in $D\\times(0,\\infty)$, with $u|_\\Gamma=f$ and zero initial data. A Laguerre-transform-based semi-discretization in time yields a sequence of stationary Navier problems $\\Delta^*u_n-\\kappa^2u_n=\\sum_{m=0}^{n-1}\\beta_{n-m}u_m$ coupled through $\\beta_n=\\kappa^2(n+1)$ and $u_n|_\\Gamma=f_n$. These are solved with a boundary integral equation framework using explicit fundamental solutions $E_n$ and layer potentials, leading to first-kind and second-kind systems for densities $q_n$ on $\\Gamma$, with well-posedness in Hölder spaces. A fully discrete scheme employing trig quadrature and a spectral-type Laguerre expansion demonstrates exponential convergence for analytic data and accurate exterior elastodynamics simulations without domain truncation.

Abstract

A numerical method for the Dirichlet initial boundary value problem for the elastic equation in the exterior and unbounded region of a smooth closed simply connected 2-dimensional domain, is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and a boundary integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the time-depended problem to a sequence of stationary boundary value problems, which are solved by a boundary layer approach resulting to a sequence of boundary integral equations of the first kind. The numerical discretization and solution are obtained by a trigonometrical quadrature method. Numerical results are included.

Figures (2)

• Figure 1: The boundary curve $\Gamma$, the source point $z\in {\rm I\! R}^2 \setminus D,$ and the measurement point $y\in D.$
• Figure 2: The $L^2$ norm of the difference between the computed and the exact solutions in logarithmic scale. In the left picture we see the convergence for the values of Table \ref{['table1']} and in the right of the Table \ref{['table2']}.

Theorems & Definitions (7)

• Theorem 2.1
• Theorem 2.2
• Definition 3.1
• Lemma 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5