Detection of a two-dimensional moving cavity

Abstract

The work here provides the mathematical and numerical groundwork for a potential thermography technique which can be used to track the time-dependent development of crystalline deposits. We develop a meshless numerical method, the method of fundamental solutions for solving the two-dimensional time-dependent heat equation, to locate an internal moving boundary ∂ D(t), where D(t) is assumed to be a perfectly insulating cavity. The inverse problem is non-linear and therefore, a least-squares minimization routine within MATLAB, FMINCON is employed to reconstruct the cavity, using known Dirichlet and Neumann data on the outer fixed boundary. The problem presented is also ill posed and, therefore, a Tikhonov regularization technique is employed.

Keywords:

• inverse problem
• thermography
• method of fundamental solutions
• non-linear optimization
• shape identification

2010 AMS Subject Classifications :

• 65N35
• 65N38
• 31A25
• 80A20

Acknowledgements

The authors thank the EPSRC and National Nuclear Laboratory for continual funding and support.