Dimension reduction of thermoelectric properties using barycentric polynomial interpolation at Chebyshev nodes

Abstract

The thermoelectric properties (TEPs), consisting of Seebeck coefficient, electrical resistivity and thermal conductivity, are infinite-dimensional vectors because they depend on temperature. Accordingly, a projection of them into a finite-dimensional space is inevitable for use in computers. In this paper, as a dimension reduction method, we validate the use of high-order polynomial interpolation of TEPs at Chebyshev nodes of the second kind. To avoid the numerical instability of high order Lagrange polynomial interpolation, we use the barycentric formula. The numerical tests on 276 sets of published TEPs show at least 8 nodes are recommended to preserve the positivity of electrical resistivity and thermal conductivity. With 11 nodes, the interpolation causes about 2% error in TEPs and only 0.4% error in thermoelectric generator module performance. The robustness of our method against noise in TEPs is also tested; as the relative error caused by the interpolation of TEPs is almost the same as the relative size of noise, the interpolation does not cause unnecessarily high oscillation at unsampled points. The accuracy and robustness of the interpolation indicate digitizing infinite-dimensional univariate material data is practicable with tens or less data points. Furthermore, since a large interpolation error comes from a drastic change of data, the interpolation can be used to detect an anomaly such as a phase transition.