Finite elements for the space approximation of a differential model for salts crystallization

TL;DR

The paper advances salt-crystallization modeling in porous media by extending a one-dimensional framework to multidimensional domains using a finite-element space discretization with implicit–explicit time stepping. It analyzes stability and convergence, conducts a sensitivity study on critical parameters in 1D, and demonstrates 2D and 3D simulations that couple moisture transport, ion migration, crystal growth, and porosity evolution, including realistic geometries. A calibration of the top-boundary exchange coefficient $K_w$ aligns the 3D FEM results with the established 1D reference dynamics, and the work provides a basis for future mechanics-informed extensions and integration into architectural material assessment workflows. Overall, the approach yields a robust, higher-dimensional tool for predicting salt-induced deterioration in porous artifacts and offers insights into parameter influence on porosity and crystallization dynamics; it also suggests a pathway toward more comprehensive, workflow-ready simulations in heritage conservation contexts.

Abstract

This article investigates a space-time differential model related to the degradation of stone artifacts caused by exposure to air and atmospheric agents, which specifically lead to the accumulation of salt crystals in the material. A numerical method based on finite-element space discretization and implicit-explicit time marching is proposed as an extension of a one-dimensional finite-difference framework introduced in the literature. Within the same one-dimensional setting, a sensitivity analysis is performed, based on the techniques developed therein. They are also used as a comparison tool for the finite-element formulation, here introduced for more realistic simulations in higher space dimensions. Considerations about stability will be provided, together with an experimental convergence analysis highlighting the performance of the proposed approach. Numerical results in two and three space dimensions, obtained by an efficient code implementation, will be presented and discussed.

Figures (15)

• Figure 1: Geometrical representation of the domain $\Omega$ for $d=1,2,3$, together with a schematic representation of the partition set on the boundary $\Gamma$.
• Figure 2: Schematic geometrical representation of the imbibition (left) and drying (right) phases.
• Figure 3: Results of the local sensitivity analysis. Relative changes, with respect to the baseline parameters, in the average porosity (top) and in the average crystallized salt (bottom) at the final time. Each plot illustrates the combined effect of perturbations in $\gamma$, $K_s$, and $K_w$. (Distinct colormaps and scales are used for the two panels).
• Figure 4: Results of the one-factor-at-a-time (OAT) sensitivity analysis.
• Figure 5: Comparison between 2D finite elements (FEM) and 1D finite difference (FD): the approximation of the quantities $\theta_l,\:c_i,\:c_s$ and $n$ is shown, for both the employed techniques, at the final instant of water absorption. FEM results have been respectively obtained from the implementation of scheme \ref{['theta k']}-\ref{['ci k']} and fixing the vertical variable $x_3$ in the range $[0\,\textrm{cm},5.85\,\textrm{cm}]$ determined by the height of the bar.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3