Perturbative Analytical Framework for Thermal Wave Diffusion in Non-linear Building Envelopes

TL;DR

This paper presents a frequency-domain analytical framework using the Riccati equation for thermal wave diffusion in non-linear building envelopes, improving computational efficiency and stability for real-time MPC applications.

Contribution

It introduces a meshless, perturbative approach that analytically handles property gradients and radiative boundaries, avoiding spatial truncation errors and state-space inflation.

Findings

Reduces peak heating load deviations by 21.9% in wetted media

Mitigates artificial nocturnal cooling fluxes of 12.0 W/m²

Maintains O(N) spatial complexity for high-speed MPC

Abstract

Model Predictive Control (MPC) in building energy management requires transient thermal models balancing thermodynamic accuracy with computational efficiency. Standard spatial discretization triggers state-space inflation, paralyzing real-time solvers, while analytical Transfer Matrix Methods (TMM) suffer from high-frequency numerical overflow and assume material homogeneity. This paper introduces a frequency-domain framework based on the continuous spatial Riccati equation. A recursive admittance mapping strictly bounds exponential growth, preventing numerical instability. Regular perturbation theory analytically resolves continuous spatial property gradients ($\lambda$(x)) and non-linear T 4 radiative boundaries as equivalent harmonic source terms. This meshless approach eliminates spatial truncation errors. It analytically corrects peak heating load deviations of 21.9% in wetted media and mitigates artificial nocturnal cooling fluxes of 12.0 W/m 2 . Preserving an O(N ) spatial complexity, the framework structurally avoids state-space inflation, ensuring the high-speed execution demanded by multi-week MPC optimization.