Time-FractionalModelofChlorideDiffusion inConcrete:Analysis Using Meshless Method

Chloride diffusion is the major factor that affects the life of concrete structures. )e time-fractional order equation can be used to describe anomalous diffusion in reinforced concrete. In this work, a time-fractional model of chloride diffusion is solved via the meshless method. )e Element-Free Galerkin (EFG) meshless method does not require meshing. One-dimensional and two-dimensional numerical examples are presented. Numerical results are in good agreement with theoretical solutions. )e initiation time of corrosion is predicted in the presented model. Simulation results are compared with experimental data. )e good agreement between EFG and experimental data indicates that time-fractional chloride diffusion in concrete can be modeled effectively by using the EFG method. )is method is beneficial for further research on anomalous chloride diffusion in concrete.


Introduction
Reinforced concrete (RC) structures in coastal areas may suffer from rebar corrosion, and chloride ingression is the main factor that affects the durability of concrete structures [1][2][3][4]. Chloride diffusion can be described by traditional models via the diffusion equation that conforms to Fick's law under the assumption that the concrete is a homogeneous and porous material [5]. However, the concrete material is actually not uniform and stable; some observers have found that the actual chloride diffusion process does not satisfy the traditional diffusion model and the chloride diffusion process particularly exhibits time-dependent feature in engineering practice, which is called anomalous diffusion [6,7]. e chloride diffusion mechanism in concrete is highly complicated because of wicking, chloride binding, permeation, and absorption, Furthermore, this case is an anomalous diffusion process, in which particles diffuse slower than the normal diffusion and its time-dependent features cannot be accurately reflected by traditional models [8].
Anomalous diffusion has sparked wide interest in the fields of physics and engineering in recent years [9]. Anomalous diffusion process is essentially a process with memory in time and nonlocality in space, the definition of the integral derivative limit is localized. Moreover, traditional diffusion equations can no longer accurately describe anomalous diffusion behaviors. Fractional-order equations are mathematical models that have been proved to accurately reflect this type of anomalous diffusion phenomenon.
Fractional differential equations are a generalization of integer-order differential equations, and their application prospect is broad. Fractional complete equations have been used in viscoelasticity, dynamics, leaky pools, cybernetics, and statistics in recent decades [10,11]. However, they are not widely applied in modeling chloride diffusion in concrete. e time-fractional diffusion model can be applied to predict chloride diffusion. It is solved through the finite difference method. At present, the finite element method (FEM) and the finite difference method (FDE) are commonly used for the numerical solution of fractional diffusion equations [12,13]. FEM or FDE must be divided into grids during solution. Grid meshing will require considerable calculation time when a structure is highly complicated.
Numerous meshless methods that avoid meshing have been developed. In recent years, these methods have been widely used in different areas, such as elasticity, thermal conduction, and fracture mechanics. e meshless method has been used to model chloride diffusion in concrete only by [14][15][16][17]. Among these meshless methods, the Element-Free Galerkin (EFG) meshless method has been used successfully to solve the problems of plates and shells [18,19], fracture mechanics and wave propagation [20][21][22][23], and electromagnetic field analysis [24]. However, this method has not been used for the time-fractional model analysis of concrete.
In the present work, the EFG meshless method is used to solve the time-fractional model of chloride diffusion in concrete. e EFG meshless method does not require element connectivity, and integration over the solution domain requires only the simple integration of cells to obtain the solution.
e governing differential equation is obtained using the Galerkin weak form, and the moving least-square (MLS) approximation is adopted in EFG [25]. e essential boundary conditions to be imposed are directly approached as a penalty in this work. e structure of this paper is as follows: in Section 2, the MLS approximation is discussed briefly. e discrete timefractional chloride diffusion model is presented in EFG form in Section 3, one-dimensional (1D) and two-dimensional (2D) numerical examples are then solved, discussed, and analyzed, and EFG and experimental results are compared in Section 4. e conclusion is presented in the following section.

MLS Approximants
e unknown concentration function C(x) is approximated by the MLS approximant C h (x). It can be written as follows [26]: where m is the number of terms in the basis, p k (m) is the monomial base function, and a k (x) is an unknown coefficient. e coefficient is obtained by minimizing function J and is written as follows: where w i (x − x i ) is a weight function. It is important in the EFG method. e weight function w i (x − x i ) is nonzero over a small neighborhood of x i , which is called the domain of influence of node.
In accordance with equations (1) and (4), we have

Time-Fractional Chloride Diffusion Model in Concrete.
Chloride diffusion in concrete is assumed to follow Fick's second law. e equation is written as where C � C (x, y, t) represents concentration; t is time; and D x and D y are diffusion coefficients in the x and y directions, respectively. When D is a constant in one dimension, the analytical solution of equation (6) is expressed as where C s is the chloride concentration at the exposure surface and erf (.) is the error function. We fully use the time-fractional diffusion equation to establish the time-fractional model of chloride ion diffusion in concrete as follows: where D αx and D αy are the diffusion coefficients in the timefractional model of chloride ion diffusion in concrete.

Time-Fractional Derivative Approximation.
Defining Δt as the time interval and t k � nΔt, (n � 1, 2, 3...), the timefractional derivative at t � t n+1 can be approximated as where R 1 and R 2 are the truncation errors and Γ(·) is the gamma function. And,

Space Discretization in EFG Meshless
Method. e weak form of equation (6) is obtained via variation as follows: By using the penalty function method to address the boundary conditions in this work, equation (14) is obtained as where alfa is the penalty parameter and equals 10 3 -10 8 .
In accordance with equation (5), we can obtain the following expressions: where Φ is the shape function and is written as follows: Substituting the above three equations into equation (14), we obtain where In accordance with the arbitrariness of δC T , via the time central difference method, then Simplifying equation (19) yields equation (20) that is the discrete expression of the time-fractional model: where

Numerical Examples
e application of the time-fractional model of the EFG meshless method is illustrated using 1D and 2D examples of chloride diffusion in concrete. MATLAB codes are developed to obtain the EFG meshless method.

1D Chloride Diffusion in Concrete.
e example is a 0.15 m × 0.15 m concrete block. e left boundary is 5% (chloride concentration). In the first example, the diffusion coefficient is assumed to be constant D α � 1.5768 × 10 − 12 m 2 /s, and in the second example, D is a time-dependent function D � D 0 (t 0 ) m , where D 0 is the diffusion coefficient at reference time, t 0 is reference time, and m is an age factor. e initial chloride concentration is zero. When α � 1, the time-fractional chloride (TFC) diffusion model in concrete is the same as the traditional chloride (TC) diffusion model. For the purpose of convergence studies, the root mean square (RMS) error is defined as R � (1/N) where N is the number of nodes, C(x i ) is the calculation with meshless methods, and C E (x i ) denotes the analytical solution.

Example 1.
e block is divided into 31 nodes. When α � 1, the EFG meshless method in the TFC model, based on the weighted least-squares (MWLS) method [17] and FEM method in the TC model, is adopted. C20e and C50e are analytical solutions according to equation (7), C20 m and C50 m denote chloride concentration in the TFC model, C20 mw and C50 mw mean chloride concentration from MWLS at exposure times of 20 and 50 years, and C20f and C50f denote chloride concentration from FEM in Figure 1. e results of the TFC model and TC model are consistent with the analytical values presented in Figure 1. e RMS error is 0.9543% in TFC, RMS error of MWLS is 1.2984%, and RMS error of FEM is 0.9869% in the TC model when t � 20 years. e errors between the simulated results and the analytical results are very small. is result shows that the simulation of the TFC model using the meshless method is correct. Figure 2 shows the change in chloride concentration at different depths with the time-fractional order α. Chloride concentration gradually decreases with the increment in depth, as shown in Figure 2. e chloride concentration at the depth of 5 cm is 0.4502%, 1.0196%, and 1.901% when the value of α is 0.5, 0.7, and 0.9, respectively. e chlorine concentration at the same depth when t � 50 years increases with the increase in α. Large values of α indicate strong chlorine permeability and correspond to high chloride concentration.
Critical chloride ion content (C t ) is selected as 0.5%, and a rebar is assumed to exist at a depth of 5 cm. e initiation period of corrosion for the start of the rebar is shown in Table 1. e predictions for the initiation periods of corrosion are clearly different. ese results show that chloride concentration increases with the increase in α. e results are almost the same when α � 1.0 in TFC and the analytical solution in TC.

Example 2.
e coefficient efficient is a time-function D � D 0 (t 0 ) m .
In this example, D 0 � 1.5768 × 10 − 12 m 2 /s, t 0 � 28 days and m � 0.1. It means D α � D 0 (t 0 ) m Γ(α) and α � 1 − m in the TFC model according to reference [8]. e parameters D α and α can be obtained by mean square displacement (MSD) and some data fitting techniques, such as least-square fitting of experimental data [27]. e values of the remaining parameters are the same as in Example 1.
As shown in Figure 3, the change trend of chlorine concentration is decreasing with depth increasing. e chloride concentration of the same depth is increasing with the increase of the α value when t � 20 years. e greater the value of α, the greater the concentration of chloride. Figure 4   e regular distribution of 31 × 31 nodes is selected in the Ω domain, and when the diffusion coefficient D is a constant, the analytical solution is as follows [28]:   e chloride concentration at the depth of 75 mm when α � 1 in the TFC model and t is 10 years is shown in Table 2. As shown in Table 2, the error between the simulated value and the theoretical value is small when the depth is 75 mm.

2D Chloride Diffusion in Concrete.
is result confirms the correctness of the simulation method. Figure 5 shows the distribution of chloride concentration when α is a different value in the TFC model and t is 20 years. As Figure 5 shows, in line with the actual situation, the numerical results have the same diffusion tendency. e chloride concentration value at the same position increases as α increases. e initiation period for the start of a rebar when C t � 0.1% at a depth of 75 mm is shown in Table 3. e initiation period is the same as the theoretical value when α � 1.0. However, the initiation period is different when a is different. Similar to the initiation period for the 1D example, the initiation period is large when α is small.

Example 4.
e coefficient efficient is a time-function D � D 0 (t 0 ) m , the left and bottom boundaries are 3%, and other conditions are the same as in Example 3. D 0 � 9.38 × 10 − 12 m 2 /s, t 0 � 28 days and m � 0.2. It means D α � D 0 (t 0 ) m Γ(α) and α � 0.8 in the TFC model. Figure 6 is the

Experimental Numerical Simulation Using TFC Model.
To further verify the correctness of the proposed model, we conducted a comparison between the simulated values and the experimental values in references [29]. e simulated    [29]. We choose the C3 (cement CEM I 42.5 N/SR3/NA) sample, surface chloride mass density is 3 kg/m 3 , and the chloride diffusion coefficient D � 5.2 × 10 − 12 m 2 /s at t � 90 days and D � 4.73 × 10 − 12 m 2 /s at t � 180 days. α in the TFC model can be obtained based on the experimental data by the least-square fitting. α is 0.90 and 0.95, when t � 90 days and t � 180 days in the TFC model.
In Figure 7, the numerical simulation of TFC and TC models has the similar diffusion tendency on the depth range. Although the results of these two models are close to experimental results, the simulated values of the TFC model are closer to the experimental data.
We compare the experimental and simulated chlorine concentration values to further verify the correctness of the present TFC model. A series of experiments with three mixtures (OPC only, either 30% fly ash or 70% slag as a partial replacement for Portland cement) on chloride ion penetration were carried out in [30]. We adopt the TFC model to simulate the experimental values of OPC concrete specimens at t � 3 years and t � 8 years. e diffusion coefficient D � 3.7178 × 10 − 11 m 2 /s, the surface concentration   Advances in Materials Science and Engineering C s � 0.35%, and α � 0.9 in the TFC model. ese parameter values are the same as the values in [8]. As shown in Figure 8, the numerical simulation is close to experimental results.

Conclusions
We applied the EFG meshless method to analyze the TFC diffusion model in RC. e numerical results from 1D and 2D prove that the TFC model is effective and show that the EFG meshless method is accurate and efficient. Chloride concentration drastically varies with the fractional order α in the TFC model. Chloride concentration at the same depth is high when α is high, and the initiation period of corrosion is prolonged with the increase in α. Notably, the time-fractional diffusion equation can also be related to the fractal of the concrete microstructure, and the corresponding fractional diffusion equation can be obtained.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.