Thermal characterization of materials using Karhunen–Loève decomposition techniques – Part I. Orthotropic materials

Abstract

A new method for reliable thermal characterization of orthotropic, homogeneous materials is proposed. It is based on the use of Karhunen-Loe`ve Decomposition (KLD) techniques in association with infrared thermography experiments or any other kind of experimental device providing dense data in the spatial coordinate. Main problem addressed in this paper is how to deal efficiently with large amount of rather noised experimental data. It is proven that orthogonal properties of KLD eigenfunctions and states allow achieving simple estimates of thermal diffusivities which depend only on the first and the second KLD eigenelements. This means that the 2D KLD approximation of the temperature field provides information enough for estimation purposes. As a result, a significant amplification of the signal/noise ratios is reached. Moreover, we prove that spatially uncorrelated noise has no effect on KLD eigenfunctions, the noise being entirely reported on states (time-dependent projection coefficients). This is particularly interesting because thermal diffusivities estimation involves spatial derivatives of the eigenfunctions calculation. Consequently, the proposed method results in an attractive combination of parsimony and robustness to noise. Indeed, because it does not require analytical solutions of the associated heat conduction problem, the method could be extended to application involving heterogeneous materials. The second part of the paper deals with this extension.

Thermal characterization of materials using Karhunen–Loève decomposition techniques – Part II. Heterogeneous materials

“…. , 2 6 as well as the corresponding contribution to the total energy of TðtÞ signals [13]: 2 i =ðAE m¼1,...,n 2 m Þ. It can be noticed that most part of the TðtÞ signals energy is captured by the two first KLD components.…”

“…These differences (residuals in the following) are due to a not strictly diagonal W " matrix. As shown in [13], the statistical correlation of an arbitrary element of eðtÞ, say " i ðtÞ, with elements " j¼1,...,n ðtÞ ( j 6 ¼ i) is almost zero, but no zero. Consequently, energy matrix W " is diagonally dominant, but not strictly diagonal.…”

“…More detailed information on the KLD of the temperature field, as well as proofs of its properties, is provided in the first part of this paper [13].…”

“…As for homogeneous materials in the first part of the paper [13], the method is based on the use of KarhunenLoe`ve (KLD) techniques in association with one single infrared thermography experiment. Compared to the inverse techniques discussed above, the proposed KLD-based method allows dealing efficiently with large amount of noise-corrupted temperature data, as those usually provided by IRT experiments.…”

“…However, maximum spatial resolution of IRT is around 20 mm. Concerning materials thermal characterization by IRT, most part of the work carried out relates to homogeneous materials [8][9][10][11][12][13] as well as to inverse methods based on simple analytical solutions of the heat conduction problem [8][9][10][11]. Thermal characterization of heterogeneous materials is obviously more complicated because involving estimation of as many thermal diffusivities as homogeneous phases (composites) or continuous spatial variations of thermal properties (functionally graded materials).…”