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

This paper proposes and validates a finite-element space discretization combined with implicit-explicit time marching to simulate salt crystallization in stone artifacts, extending previous one-dimensional models to higher dimensions while conducting sensitivity analysis, stability considerations, and convergence testing.

The Gist

Imagine a historic stone building, like an ancient temple or a Renaissance church. Over centuries, it faces a silent enemy: salt.

When rain soaks into the stone, it carries dissolved salt deep inside the tiny pores (holes) of the rock. When the sun comes out and the stone dries, that water evaporates, but the salt stays behind. It crystallizes, growing like tiny, sharp ice crystals inside the stone's pores. Just like ice expanding in a water pipe, these growing crystals push against the stone walls, causing it to crack, flake, and crumble. This is the slow, painful death of our cultural heritage.

This paper is about building a super-smart digital simulator to predict exactly how and where this damage happens, so we can protect these treasures.

Here is the breakdown of their work, using simple analogies:

1. The Old Map vs. The New GPS

Previously, scientists had a map to track this salt damage, but it was a one-dimensional map (like a flat line). It could tell you what happens if you look at a stone from the bottom to the top, but it couldn't see what happens on the sides or in the corners. Real stones are 3D objects, and salt doesn't just move up and down; it spreads out in all directions.

The authors took that old, flat map and upgraded it to a 3D GPS. They built a complex mathematical model that can simulate a whole block of stone in 3D space (length, width, and height), not just a single line.

2. The Four Characters in the Story

To simulate the stone, the computer tracks four main "characters" that change over time:

• The Water (Moisture): The liquid soaking into the stone.
• The Salt Ions: The invisible salt particles floating in the water.
• The Salt Crystals: The solid, damaging rocks forming inside the holes.
• The Porosity: The amount of "empty space" (holes) in the stone. As crystals grow, they fill up these holes, making the stone tighter and more brittle.

The magic of this model is that these characters are best friends who can't live without each other. The water moves the salt; the salt turns into crystals; the crystals block the holes; and the blocked holes change how the water moves. It's a chaotic dance, and the math has to keep everyone in sync.

3. The Two-Act Play: Soaking and Drying

The simulation mimics a real-life experiment in two acts:

• Act 1: The Soak (Imbibition). Imagine dipping the bottom of a sponge into salty water. The water rushes up, carrying salt with it. The model tracks how fast the water rises and where the salt starts to pile up.
• Act 2: The Dry. Now, imagine taking that wet sponge out and letting it sit in the sun. The water evaporates from the top and sides, but the salt stays trapped inside. This is when the crystals grow and the damage begins.

4. The New Tool: Finite Elements (The LEGO Approach)

To solve the math for a 3D stone, the authors used a method called Finite Element Method (FEM).

• The Old Way (Finite Differences): Imagine trying to measure a curved mountain using only a ruler. You have to force the curve into a jagged, stair-step shape. It works okay for simple lines, but it gets messy and inaccurate for complex shapes.
• The New Way (FEM): Imagine building the mountain out of thousands of tiny, flexible LEGO bricks. You can mold these bricks to fit any curve, any corner, and any weird shape of a real stone artifact. This allows the computer to calculate the salt movement with much higher precision, even on complex, real-world statues.

5. Testing the Engine

Before trusting the new 3D simulator, the authors had to make sure it was reliable:

• Sensitivity Analysis (The "What If" Game): They asked, "What if the salt grows 10% faster? What if the air is 10% drier?" They found that the model is robust. Small changes in the weather or salt type don't cause the simulation to crash or give wild, wrong answers. It behaves like real stone does.
• Convergence Check: They compared their new 3D LEGO method against the old 1D ruler method. The new method was not only more accurate but also more stable, proving it's the better tool for the job.

6. The Results

They ran the simulation on a digital "brick" in 2D and 3D.

• They watched the "water front" race up the stone.
• They saw the salt crystals slowly accumulate, mostly near the top where the water evaporates (the "salt line").
• They confirmed that the stone's internal structure (porosity) gets clogged up, just like a clogged sink, which slows down future water movement.

Why Does This Matter?

This isn't just about math; it's about saving history.
By having a tool that can accurately predict where salt damage will happen in a 3D statue, conservators can:

1. Know exactly where to apply protective treatments.
2. Understand why some parts of a building are crumbling faster than others.
3. Test different conservation strategies in the computer before touching the real stone.

In short, the authors built a digital crystal ball that helps us see the invisible battle between water, salt, and stone, giving us a fighting chance to preserve our cultural heritage for the future.

Technical Summary

1. Problem Statement

The paper addresses the degradation of stone artifacts in cultural heritage caused by salt crystallization. This phenomenon occurs when porous materials (like stone and brick) absorb saline water, leading to the transport of dissolved ions, their precipitation within the pore network, and the subsequent reduction of porosity. This process induces mechanical stress and structural deterioration.

While existing mathematical models (specifically reference [5]) effectively describe these coupled processes (moisture transport, salt migration, and crystallization), they are limited to one-dimensional (1D) spatial representations. This limitation restricts their applicability to realistic, complex geometries found in actual heritage structures. The authors aim to extend this model to multi-dimensional domains (2D and 3D) to enable more realistic simulations of diffusion-driven phenomena and to provide a robust numerical framework for analyzing salt-induced damage.

2. Mathematical Model

The study builds upon the differential model introduced in [5], describing a system of nonlinear, time-dependent partial differential equations (PDEs) with strong coupling between transport, reaction, and microstructural evolution.

State Variables:

• $\theta_l(x,t)$: Volumetric fraction of liquid water.
• $c_i(x,t)$: Concentration of dissolved salt ions.
• $c_s(x,t)$: Concentration of crystallized salt.
• $n(x,t)$: Porosity of the material (coupled to $c_s$ via $n = n_0 - \gamma c_s$).

Governing Mechanisms:

1. Moisture Transport: Modeled as a nonlinear diffusion process driven by capillary effects, governed by a function $B(s)$ that relates saturation to hydraulic conductivity.
2. Salt Transport: Governed by a convection-diffusion equation where advection is induced by moisture flux and diffusion by concentration gradients.
3. Crystallization Kinetics: Modeled as a sink term for dissolved ions and a source for solid crystals, dependent on local ion concentration and available pore space.

Phases:
The simulation is divided into two phases:

• Imbibition: Water absorption from a saline source at the bottom, with evaporation at the top.
• Drying: Evaporation without further salt supply, simulating the drying cycle.

3. Methodology

Numerical Discretization

The authors propose a hybrid space-time discretization:

• Time Discretization: An Implicit-Explicit (IMEX) scheme using first-order finite differences. This allows for stable handling of stiff terms while maintaining computational efficiency.
• Space Discretization: A Finite Element Method (FEM) approach using piecewise linear elements. This is the core innovation, replacing the 1D Finite Difference (FD) method to handle complex 2D and 3D geometries.
• Implementation: The solver is implemented using the FEniCS computing platform.

Sensitivity Analysis

Before extending to higher dimensions, a local sensitivity analysis was performed on the 1D model to assess the robustness of the system against perturbations in three key parameters:

1. $\gamma$: Specific volume of crystals.
2. $K_s$: Crystallization rate coefficient.
3. $K_w$: Evaporation/exchange coefficient at the boundary.
   The analysis utilized a "neighborhood" approach (varying multiple parameters simultaneously) rather than a simple one-factor-at-a-time (OAT) approach, evaluating the impact on average porosity and total crystallized salt.

Boundary Conditions

• Imbibition: Dirichlet conditions at the bottom (saturated moisture/salt); Robin conditions at the top (evaporation); Neumann (no-flux) on lateral sides for $d > 1$.
• Drying: Dirichlet (zero moisture) at top/bottom; Neumann (no-flux) for salt ions everywhere.

4. Key Contributions

1. Multidimensional Extension: Successfully extended the 1D salt crystallization model to 2D and 3D domains, enabling the simulation of realistic prismatic geometries.
2. FEM Formulation: Developed a stable and convergent FEM-based solver that handles the nonlinear coupling of moisture transport and crystallization in higher dimensions, overcoming the geometric limitations of previous FD schemes.
3. Sensitivity Analysis: Demonstrated that the model is robust against parameter variations (up to $\pm 10\%$), confirming that parameters calibrated in 1D studies remain valid for multidimensional simulations.
4. Convergence Analysis: Conducted an experimental convergence study showing that the FEM approach achieves $O(\Delta t)$ in time and $O(h^2)$ in space (for linear elements) for most variables, outperforming the traditional FD scheme in terms of convergence rates for this specific nonlinear problem.

5. Numerical Results

The authors presented simulations in 1D, 2D, and 3D:

• 2D Bar (Long Imbibition): Simulated over ~4444 hours. Results showed complete moistening of the bar and a localized porosity reduction of ~1.6% at the top due to salt deposition. The FEM results aligned closely with the 1D FD benchmark.
• 2D Bar (Imbibition + Drying): Simulated a full cycle. The drying phase successfully reduced moisture content without significantly altering the already deposited salt mass or porosity.
• 3D Bar (Realistic Experiment): Simulated a prism ($0.3 \times 0.3 \times 0.75$ cm). The 3D plots visualized the temporal evolution of water absorption and crystal growth.
  • Observation: Water absorption ($\theta_l$) occurred significantly faster than crystal deposition ($c_s$).
  • Porosity Change: In the short 3D experiment, the global porosity change was negligible (~0.00001%), attributed to the short duration of water supply. The authors note that longer experiments would yield more significant porosity reduction.

6. Significance and Future Work

• Practical Application: The proposed FEM framework provides a predictive tool for analyzing salt-induced degradation in complex, real-world heritage structures, moving beyond simplified 1D assumptions.
• Computational Efficiency: The IMEX-FEM scheme offers a stable and efficient method for simulating long-term degradation processes in 3D.
• Future Directions: The authors plan to extend the model to include mechanical damage (mass loss and structural failure) resulting from repeated crystallization cycles. They also aim to integrate this algorithm into the Stoneverse platform for broader application in cultural heritage conservation.

In conclusion, this work bridges the gap between theoretical 1D modeling and practical 3D simulation, providing a robust numerical tool for understanding and mitigating salt crystallization damage in porous building materials.