Estimating eigenvalues of an anisotropic thermal tensor from transient thermal probe measurements

Steve Rosencrans; Xuefeng Wang; Shan Zhao

doi:10.3934/dcds.2013.33.5441

Article Contents

2013, Volume 33, Issue 11&12: 5441-5455. Doi: 10.3934/dcds.2013.33.5441

This issue Previous Article Integration with vector valued measures Next Article Hardy type inequalities and hidden energies

Estimating eigenvalues of an anisotropic thermal tensor from transient thermal probe measurements

1.
Department of Mathematics, Tulane University, New Orleans, LA 70118
2.
Mathematics Department, Tulane University, New Orleans, LA 70118
3.
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350

Received: July 31, 2011

Revised: February 29, 2012

Published: October 2013

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Abstract

We propose a new method for estimating the eigenvalues of the thermal tensor of an anisotropically heat-conducting material, from transient thermal probe measurements of a heated thin cylinder.
We assume the principal axes of the thermal tensor to have been identified, and that the cylinder is oriented parallel to one of these axes (but we outline what is needed to overcome this limitation). The method involves estimating the first two Dirichlet eigenvalues (exponential decay rates) from transient thermal probe data. These implicitly determine the thermal diffusion coefficients (thermal tensor eigenvalues) in the directions of the other two axes. The process is repeated two more times with cylinders parallel to each of the remaining axes.
The method is tested by simulating a temperature probe time-series (obtained by solving the anisotropic heat equation numerically) and comparing the computed thermal tensor eigenvalues with their true values. The results are generally accurate to less than $1\%$ error.

Keywords:

Thermal tensor,
inverse problem.

Mathematics Subject Classification: Primary: 35K05, 80-04, 80A23.

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