Optimizing climate data integration for thermal properties characterization in building walls: A comparative investigation of fitting techniques and inversion algorithms using thermal quadrupoles

Solving the heat equation is fundamental in various applications, with methods ranging from analytical solutions under simplified assumptions to numerical approaches for complex scenarios. The thermal quadrupole method offers a unique advantage by transforming the heat equation into a linear system using Laplace and integral transforms (for time and space domains), providing efficient solutions for multilayered building walls. This work incorporates real climate data—hourly external temperatures and solar heat flux—into the quadrupole model, where the wall is represented as a 2D axisymmetric cylinder, necessitating Hankel transforms for spatial resolution in the radial direction. Climate data, characterized by significant fluctuations, are fitted for transformation using either Fourier series fitting or a segmented discrete Laplace transform method. After solving in the transformed domain, inversion to the original time-space domain is performed using Stehfest, De Hoog, and Den Iseger algorithms. A parametric analysis evaluates the performance of various fitting and inversion combinations. Results are compared with finite element solutions (COMSOL), emphasizing computational efficiency and accuracy. In addition to this cylindrical multilayer wall application, the study extends to testing fitting techniques and inversion algorithms on a 1D problem. This problem, solved with quadrupoles and through FreeFEM++, provides a baseline for assessing the robustness of the proposed methodologies. The comparative study highlights the reliability of the combined fitting and inversion approaches under diverse problem settings. This work is conducted within the framework of the RESBIOBAT project funded by the French national research agency (under grant ANR-21-CE22-0018).

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