Reconstruction of an unknown cavity with Robin boundary condition inside a heat conductor

TL;DR

This paper develops a mathematical and computational approach to reconstruct an unknown cavity with Robin boundary conditions inside a heat conductor using boundary measurements from active thermography, with proven asymptotic analysis and reconstruction scheme.

Contribution

It introduces a linear sampling method for Robin cavity reconstruction and provides mathematical justification and asymptotic analysis for the approach.

Findings

Successfully reconstructs cavity boundary and distance from boundary measurements.

Provides mathematical justification for the linear sampling method.

Analyzes the indicator function's asymptotic behavior.

Abstract

Active thermography is a non-destructive testing technique to detect the internal structure of a heat conductor, which is widely applied in industrial engineering. In this paper, we consider the problem of identifying an unknown cavity with Robin boundary condition inside a heat conductor from boundary measurements. To set up the inverse problem mathematically, we first state the corresponding forward problem and show its well-posedness in an anisotropic Sobolev space by the integral equation method. Then, taking the Neumann-to-Dirichlet map as mathematically idealized measured data for the active thermography, we present a linear sampling method for reconstructing the unknown Robin-type cavity and give its mathematical justification by using the layer potential argument. In addition, we analyze the indicator function used in this method and show its pointwise asymptotic behavior by investigating the reflected solution of the fundamental solution. From our asymptotic analysis, we can establish a pointwise reconstruction scheme for the boundary of the cavity, and can also know the distance to the unknown cavity as we probe it from its inside.