Spectral Methods - Part 2: A comparative study of reduced order models for moisture transfer diffusive problems

This paper explores in details the capabilities of two model reduction techniques – the Spectral Reduced Order Model (Spectral–ROM) and the Proper Generalised Decomposition (PGD) – to numerically solve moisture diffusion problems. Both techniques assume separated tensorial representation of the solution by a finite sum of function products. The Spectral–ROM fixes a set of spatial basis functions to be the Chebyshev polynomials and then, a system of ordinary differential equations is built to compute the temporal coefficients of the solution using the Galerkin projection method, while the PGD aims at computing directly the basis of functions by minimising the residual. Both approaches are compared for three different cases: i) linear transfer; ii) parametric problems and iii) nonlinear diffusive transfer. Results have highlighted that both numerical techniques provide accurate solution and enable to reduce significantly the order of the model, allowing a fast computation of physical phenomena such as the moisture buffer effects that occur in porous building materials. For the linear and nonlinear cases, the Spectral–ROM error decreases faster than the one for the PGD. Moreover, fewer modes are required for the Spectral to compute a solution with equivalent accuracy. However, for the parametric case, the PGD computed a reduced order model whose outputs depend not only on the coordinates of space and time x and t , but also on the coordinate of the parameter belonging to a defined interval. On the other hand, the outputs of the Spectral–ROM depend only on the coordinates of space and time. The solution of the parametric problem is obtained by computing the solution for each numerical value of a given parameter within the defined interval.