Adaptive integration of element matrices in finite element moisture transfer simulations

Abstract

While serving different purposes, numerical simulations of moisture and heat transfer in soils and in building components are very similar in methodology: in both cases, spatially and temporally discretised equations for transfer of moisture and heat in porous materials are solved subject to (atmospheric) boundary conditions. The strongly non-linear transfer equations and boundary conditions however render such hygrothermal simulations computationally very expensive, and an efficient numerical solution algorithm is required. Such increasingly efficient numerical solution schemes allow for more, larger, longer or more precise simulations, widening the application capabilities of hygrothermal simulations. The computational cost of hygrothermal simulations revolves around the serial iterative com-position and decomposition of the coefficient matrix of the system of algebraic equations de-scribing the discretised moisture and heat transfer, and is thus determined by the cost of one (de)composition, and the number of required (de)compositions. This article presents two op-timisation measures for simulations of moisture and heat transfer in building components un-der atmospherical excitation: adaptive integration and variations on the Newton-Raphson iterative scheme. Adaptive integration targets the cost of one (de)composition, while the varia-tions on Newton-Raphson aims at the number of required (de)compositions. While exempli-fied by building physical simulations, the presented optimisation measures are equally valid for simulations of moisture and heat transfer in soils. It will be demonstrated that the common preference for low-order numerical integration of the finite element matrices has an adverse effect on the required spatial discretisation: a fine dis-cretisation throughout is needed for accurate simulation of the moving moisture fronts typical of infiltration problems. Adaptive integration allows to merge low-order numerical integration with rougher spatial discretisations, reducing the number of required integration points and of discretisation nodes. A second section of the article investigates the efficiency of (variations on) the Newton-Raphson scheme. It will be demonstrated that appropriate application of Newton-Raphson on the boundary conditions, of modified iteration and of separate convergence criteria can drastically diminish the number of required (de) compositions.