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

Roman Chapko, B. Tomas Johansson, Leonidas Mindrinos

Research output: Contribution to journalArticle

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 analyzed and made explicit, and standard quadrature rules are used for discretization. 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.

Original languageEnglish
Article number112463
JournalJournal of Computational and Applied Mathematics
Volume367
Early online date17 Sep 2019
DOIs
Publication statusE-pub ahead of print - 17 Sep 2019

Fingerprint

Integral Solution
Boundary integral equations
Elastodynamics
Boundary Integral
Linear systems
Lateral
Cauchy Problem
Stable Solution
Tikhonov Regularization
Quadrature Rules
Boundary Integral Equations
Hyperbolic Equations
Missing Data
Cauchy
Governing equation
Discretization
Linear Systems
Singularity
Transform
kernel

Bibliographical note

© 2019, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/

Keywords

  • Boundary integral equations
  • Cauchy problem
  • Elastodynamics
  • Laguerre transformation
  • Nyström method
  • Tikhonov regularization

Cite this

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title = "On a boundary integral solution of a lateral planar Cauchy problem in elastodynamics",
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 analyzed and made explicit, and standard quadrature rules are used for discretization. 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.",
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On a boundary integral solution of a lateral planar Cauchy problem in elastodynamics. / Chapko, Roman; Johansson, B. Tomas; Mindrinos, Leonidas.

In: Journal of Computational and Applied Mathematics, Vol. 367, 112463, 15.03.2020.

Research output: Contribution to journalArticle

TY - JOUR

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AU - Chapko, Roman

AU - Johansson, B. Tomas

AU - Mindrinos, Leonidas

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AB - 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 analyzed and made explicit, and standard quadrature rules are used for discretization. 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.

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KW - Elastodynamics

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KW - Nyström method

KW - Tikhonov regularization

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