On a boundary integral solution of a lateral planar Cauchy problem in elastodynamics

TL;DR

This paper introduces a boundary integral method for stably reconstructing missing boundary data in elastodynamics problems within annular domains, using Laguerre transforms and Tikhonov regularization to address ill-posedness.

Contribution

It presents a novel boundary integral approach combining Laguerre transforms and regularization for reconstructing boundary data in elastodynamics, improving stability and practicality.

Findings

Method effectively reconstructs missing boundary data

Numerical results demonstrate stability and accuracy

Approach applicable to practical elastodynamics problems

Abstract

A boundary integral based method for the stable reconstruction of missing boundary data is presented for the governing hyperbolic equation of elastodynamics in annular planar domains. Cauchy data in the form of the solution and traction is reconstructed on the inner boundary curve from the similar data given on the outer boundary. The ill-posed data reconstruction problem is reformulated as a sequence of boundary integral equations using the Laguerre transform with respect to time and employing a single-layer approach for the stationary problem. Singularities of the involved kernels in the integrals are analysed and made explicit, and standard quadrature rules are used for discretisation. Tikhonov regularization is employed for the stable solution of the obtained linear system. Numerical results are included showing that the outlined approach can be turned into a practical working method for finding the missing data.