Comparative investigation of numerical methods for incorporating real climate data into thermal quadrupole models for building wall applications: fitting techniques, and Laplace inversion algorithms

The thermal quadrupole method provides the advantage of expressing the partial differential formulation of the heat equation as a linear system in transformed time (Laplace transform) and space (integral transforms) domains. It allows faster computations compared to standard techniques such as Finite Element Methods. The following work concerns the incorporation of climate data recordings of hourly external temperature and solar heat flux in the thermal quadrupole method for solving the heat equation through a multilayered building wall. Two methods are proposed for the purpose of applying Laplace transforms to the discrete sets of climate data: a global Fourier series fit, accounting for severe fluctuations and peaks with the number of harmonics depending on dataset size; and a discrete Laplace transform methodology applied to a global series of linearly computed sub-series over defined intervals. Two models are investigated, a 1D heat transfer problem in Cartesian coordinates and a 2D axisymmetric representation in cylindrical coordinates, the latter dictating Hankel transforms for the space domain. After solving in the transformed domains, the challenge lies in accurately retrieving time-domain results. Three Laplace inversion algorithms—Stehfest, De Hoog, and Den Iseger—are investigated for their numerical stability, accuracy, and efficiency. A parametric analysis related to parameters of the data fitting and Laplace inversion methods is carried out. Results of different combinations of the fitting method/inversion algorithm (or a coupling of algorithms) are provided and compared with a finite element resolution of the thermal problems (FreeFEM++ and COMSOL) with an emphasis on computational time enhancements. The main objective of this work is to develop a numerically efficient direct model suitable for future application in inverse methods.