# American Institute of Mathematical Sciences

November  2013, 33(11&12): 5441-5455. doi: 10.3934/dcds.2013.33.5441

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

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

Received  August 2011 Revised  March 2012 Published  May 2013

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.
Citation: Steve Rosencrans, Xuefeng Wang, Shan Zhao. Estimating eigenvalues of an anisotropic thermal tensor from transient thermal probe measurements. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5441-5455. doi: 10.3934/dcds.2013.33.5441
##### References:
 [1] F. M. Arscott, "Periodic Differential Equations," Pergamon Press, 1964. [2] G. Backstrom and J. Chaussy, Determination of thermal conductivity tensor and heat capacity of insulating solids, J. Phys. E: Sci. Instrum., 10 (1977), 767-769. doi: 10.1088/0022-3735/10/8/003. [3] U. Brydsten and G. Backstrom, Hot strip determination of the thermal conductivity tensor and heat capacity of crystals, Int'l. J. Thermophysics, 4 (1983), 369-387. doi: 10.1007/BF01178787. [4] H. S. Carslaw and J. C. Jaeger, "Conduction of Heat in Solids," $2^\text{nd}$ edition, Oxford University Press, 1959. [5] V. Isakov, "Inverse Problems for Partial Differential Equation," Springer, 1998. [6] J. Li, S. Rosencrans, X. Wang and K. Zhang, Asymptotic analysis of a Dirichlet problem for the heat equation on a coated body, Proceedings of the American Math. Society, 137 (2009), 1711-1721. doi: 10.1090/S0002-9939-08-09766-9. [7] J. M. Macleod, Instructions for operating the divided bar apparatus for thermal conductivity measurement at the GSC, calgary, Open File 3444, Geological Survey of Canada, (1997). doi: 10.4095/209020. [8] V. Pereyra and G. Scherer, Exponential data fitting, in "Exponential Data Fitting and Its Applications," 1-26, Bentham, (2010). doi: 10.2174/97816080504821100101. [9] Per Sundqvist, Exponential curve-fitting without start-guess, http://www.mathworks.com/matlabcentral/fileexchange/21959.

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##### References:
 [1] F. M. Arscott, "Periodic Differential Equations," Pergamon Press, 1964. [2] G. Backstrom and J. Chaussy, Determination of thermal conductivity tensor and heat capacity of insulating solids, J. Phys. E: Sci. Instrum., 10 (1977), 767-769. doi: 10.1088/0022-3735/10/8/003. [3] U. Brydsten and G. Backstrom, Hot strip determination of the thermal conductivity tensor and heat capacity of crystals, Int'l. J. Thermophysics, 4 (1983), 369-387. doi: 10.1007/BF01178787. [4] H. S. Carslaw and J. C. Jaeger, "Conduction of Heat in Solids," $2^\text{nd}$ edition, Oxford University Press, 1959. [5] V. Isakov, "Inverse Problems for Partial Differential Equation," Springer, 1998. [6] J. Li, S. Rosencrans, X. Wang and K. Zhang, Asymptotic analysis of a Dirichlet problem for the heat equation on a coated body, Proceedings of the American Math. Society, 137 (2009), 1711-1721. doi: 10.1090/S0002-9939-08-09766-9. [7] J. M. Macleod, Instructions for operating the divided bar apparatus for thermal conductivity measurement at the GSC, calgary, Open File 3444, Geological Survey of Canada, (1997). doi: 10.4095/209020. [8] V. Pereyra and G. Scherer, Exponential data fitting, in "Exponential Data Fitting and Its Applications," 1-26, Bentham, (2010). doi: 10.2174/97816080504821100101. [9] Per Sundqvist, Exponential curve-fitting without start-guess, http://www.mathworks.com/matlabcentral/fileexchange/21959.
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