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

Roman Chapko, B. Tomas Johansson, Leonidas Mindrinos

TL;DR

This work develops a boundary-integral framework to stably reconstruct missing lateral Cauchy data for a 2D elastodynamic Cauchy problem in an annular domain. By applying a Laguerre transform in time and representing the stationary problems with a fundamental sequence $E_n$ via single-layer potentials, the authors reduce the time-dependent ill-posed problem to a sequence of boundary-only equations on $\Gamma_1$ and $\Gamma_2$. They perform a careful singularity analysis and kernel-splitting to enable accurate 2π-periodic Nyström discretization and employ Tikhonov regularization to stabilize the linear systems, yielding accurate reconstructions of inner-boundary data from outer-boundary measurements. Numerical experiments demonstrate stable recovery of missing data and time-dependent solutions, highlighting the method’s practical viability for elastodynamic Cauchy 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.

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

TL;DR

This work develops a boundary-integral framework to stably reconstruct missing lateral Cauchy data for a 2D elastodynamic Cauchy problem in an annular domain. By applying a Laguerre transform in time and representing the stationary problems with a fundamental sequence via single-layer potentials, the authors reduce the time-dependent ill-posed problem to a sequence of boundary-only equations on and . They perform a careful singularity analysis and kernel-splitting to enable accurate 2π-periodic Nyström discretization and employ Tikhonov regularization to stabilize the linear systems, yielding accurate reconstructions of inner-boundary data from outer-boundary measurements. Numerical experiments demonstrate stable recovery of missing data and time-dependent solutions, highlighting the method’s practical viability for elastodynamic Cauchy 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.

Paper Structure

This paper contains 12 sections, 2 theorems, 104 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Combination of the Laguerre transform and the boundary integral equation method
  3. A system of boundary integral equations for (\ref{['s_e']})-- (\ref{['s_bc']})
  4. The sequence of fundamental solutions $E_n$ in (\ref{['s_p']})
  5. Calculation of the traction of the fundamental sequence $E_n$
  6. Analysis of singularities of the kernels in the system (\ref{['sie']})
  7. Transformation to 2$\pi$-periodic integral equations
  8. Expressions for the singular kernel $H^n_{\ell,\ell}$
  9. Expressions for the singular kernel $Q_{\ell,\ell}^n$
  10. Full discretization
  11. Numerical examples
  12. Conclusion

Key Result

Theorem 2.1

A sufficiently smooth function (lag6) is the solution of the time-dependent problem (h_eq)-- (h_bc) if and only if its Fourier-Laguerre coefficients $u_n$ for $n = 0,1,\ldots$, solve the sequence of stationary problems (s_e)-- (s_bc).

Figures (2)

  • Figure 1: The boundary $\Gamma_1 \cup \Gamma_2$ of $D$, the source point $z_1 \in {\rm I\! R}^2 \setminus D$, and the measurement point $z_2 \in D$, in the first (left) and in the second (right) example.
  • Figure 2: The $L^2$-norm (logarithmic scale) of the difference between the computed (\ref{['eq_com']}) and the exact solution (\ref{['eq_exact']}). The convergence of the first component from Table \ref{['table1']} (left) and of the second component from Table \ref{['table2']} (right).

Theorems & Definitions (2)

  • Theorem 2.1
  • Theorem 2.2