Introduction (top)

Contact Information (top)

The model application (top)

In many industrial applications one wants to determine the temperature on the surface of a body, but where the surface itself is inaccessible for measurements. It may also be the case that locating a measurement device on the surface would disturb the measurements so that an incorrect temperature is recorded. In such cases one is restricted to internal measurements. The situation is illustrated in Figure 1.1.

In a one-dimensional setting, this situation can be modeled as Cauchy problem for the heat equation; where data is given along the line x=1 and the solution is sought at x=0. This is a frequently occuring situation in many remote sensing applications.

The Cauchy problem for a parabolic equation is ill-posed in the sense that small errors in the data (e.g. measured temperatures) may lead to errors of chatastrophic magnitute in the computed solution. This leads to computational challenges. However by using an appropriate regularization scheme good numerical solutions can be computed. Finding good regularization schemes is an ongoing research topic.

Figure 1.1:  Interior temperature measurements.

References (top)

1. J. V. Beck, B. Blackwell, and S. R. Clair. Inverse Heat Conduction. Ill-Posed Problems. Wiley, New York, 1985.
2. L. Eldén. Solving the sideways heat equation by a 'method of lines'. J. Heat Transfer, Trans. ASME, 119:406-412, 1997.
3. L. Eldén, F. Berntsson, and T. Reginska. Wavelet and Fourier methods for solving the sideways heat equation. Technical Report LiTH-MAT-R-97-22, Department of Mathematics, Linköping University, 1997.
4. H.A. Levine. Continuous data dependence, regularization, and a three lines theorem for the heat equation with data in a space like direction. Ann. Mat. Pura Appl. (IV), CXXXIV:267-286, 1983.

Software tools (top)

Several MATLAB routines has been written for solving the Sideways Heat Equation. These routines are written for the purpose of testing ideas, and mostly for use within our research group. But most of the code is fairly easy to understand and the online documentation is such that the code should be possible to use with a minimum of trouble. In particular there are several demos included, that are intended to demonstrate how the programs are to be used. Also there is an example where the tools are applied to a problem with actual measured data.

• A complete list of all programs written for solving the Sideways Heat equation is available. [Contents.m]
• The source code for all the programs. Only available as a compressed tar archive. [shetools.tar.Z]

Publications (top)

1. P. Wikström, W. Blasiak and F. Berntsson. Estimation of the Transient Surface Temperature, Heat Flux and Effective Heat Transfer Coefficient of a Slab in an Industrial Reheating Furnace by using an Inverse Method, Steel Research International, vol. 78, no. 1, pp. 31-38, January 2007.
2. F. Berntsson. Sequential Solution of the Sideways Heat Equation by Windowing of the Data Technical report LiTH-MAT-R-2002-6, Department of Mathematics, Linköping university, April 2002.
   
3. F. Berntsson. Numerical methods for solving a non-characteristic Cauchy problem for a parabolic equation, Technical report LiTH-MAT-R-2001-17, Department of Mathematics, Linköping university, September 2001.
   
4. F. Berntsson, and L. Eldén. An Inverse Heat Conduction problem and an Application to Heat Treatment of Aluminium.
   Presented at: International Symposium on Inverse Problems In Engineering Mechanics, Nagano, Japan, March 2000.
   
5. F. Berntsson.A Spectral Method for Solving the Sideways Heat Equation. Inverse Problems, vol. 15, pp. 891-906, August 1999. Also Tech. Report LiTH-MAT-R-99-06
   Journal.
6. F. Berntsson.Numerical Methods for an Inverse Heat Conduction Problem. Proceedings to the 10th Conference of the European Consortium for Mathematics in Industry, Göteborg, June 1998, In Progress in Industrial Mathematics at ECMI 98, pp. 240-246, B.G Teubner, 1999.
   
7. L. Eldén, F. Berntsson, and T. Reginska. Wavelet and Fourier Methods for Solving the Sideways Heat Equation.SIAM J. Sci. Comput., vol. 21, no. 6, pp. 2187-2205, 2000. Also Tech. Report LiTH-MAT-R-97-22.
   Journal.
8. F. Berntsson, L. Eldén, R. Padro´n and D. Loyd. A Comparison of Three Numerical Methods for an Inverse Heat Conduction Problem and an Industrial Application. Tenth International Conference on Numerical Methods for Thermal Problems, Swansea, July 1997
   
9. T. Reginska and L. Eldén. Solving the Sideways Heat Equation by a Wavelet-Galerkin Method. Inverse Problems, 13(1997), 1093-1106.
   Journal.
10. L. Eldén. Solving an Inverse Heat-Conduction Problem by a 'Method of Lines' J. Heat Transfer, Trans. ASME, 119(1997), 406-412.
    
11. L. Eldén.Numerical Solution of the Sideways Heat Equation by Difference Approximation in Time. Inverse Problems 11(1995), 913-923.
    Journal.
12. L. Eldén. Numerical Solution of the Sideways Heat Equation. In Inverse Problems in Diffusion Processes. Proceedings in Applied Mathematics, ed. H. Engl and W. Rundell, SIAM, Philadelphia 1995.