Numerical Analysis of Chloride Diffusion in Concrete with Time Varying Coefficient Based on the ADI Method

In this paper, a two-dimensional finite difference numerical model with time varying coefficient using Alternating Direction Implicit Method (ADI) is developed to predict Chloride diffusion in concrete. This model is proved to be unconditionally stable and has higher accuracy. And a numerical example is given to show the effectiveness of this model.


Introduction
The diffusing mechanism of Chloride ion in reinforced concrete structures is very complex. Generally, it includes diffusion effect, capillary effect, permeation effect, chemical migration effect and their combinations, and the diffusion effect plays a leading role. Model of Chloride ion diffusion equation is established mainly based on the Fick's second law [1] which can combine Chlorine ion concentration with diffusion coefficient and diffusion time. Chloride ion diffusion coefficient is often considered to be a constant when we calculate the numerical model of the Chloride ion diffusion. Considering the two-dimensional diffusion of Chloride ion under actual conditions, the equation can be rewritten as [2] 2 2 where C is the Chloride ion concentration, D is the constant diffusion coefficient. With the initial conditions 0 ( , , 0) , (0 , ) C x y C x y = < < ∞ (2) and the boundary conditions   [4][5][6][7], finite element method in [8][9], and boundary element method in [10][11] However, in the 1920s, people begin to think that the diffusion coefficient is influenced by the environment, a lot of research shows that the longer the time is, the smaller the diffusion coefficient is. Though the proof of a lot of experiments, it is considered that the diffusion coefficient can be expressed as follow [12]: where α is related to the water cement ratio and affected by itself attribute of the component and the surrounding conditions. For example,  [13], where / w c is the water cement ratio of concrete component. One-dimensional diffusion model with time varying diffusion coefficient and some modified model are discussed in [14][15][16][17][18][19][20].
Motivated by the above mentioned studies, we will consider a two-dimensional Chloride ion diffusing problem in a finite rectangle with time varying diffusion coefficient based on Fick's second law. In this paper, a two-dimensional ADI numerical model of Chloride diffusion is established in Section 2. At the same time, the truncation error and the stability are analyzed in Section 3. A numerical example is shown in Section 4, which can effectively predict the diffusion of Chloride ion in concrete. At last, in Section 5 the conclusion is given.

Establishment of ADI Model
Based on Fick's second law, a two -dimensional Chloride ion diffusing model in a finite rectangle with time varying diffusion coefficient is where ( , , ) C x y t is the Chloride ions concentration of point ( , ) x y at diffusion time t , ,

Mesh
First, the finite area

Discretion
According to the above mesh and the Taylor expansion, we have

ADI Model
According to ADI method [21], when calculating the Chloride ions concentration at  , and taking t nτ = , according to the equation, we can get The second step: And at the second half steps of time, that is, from the 1 2 n + time layer to the 1 n + time layer, using the explicit method to solve , and using the implicit method to solve , and taking t nτ = , according to the equation, we can get For simplicity, assume where ( ) Thus, a two-dimensional ADI numerical model of Chloride diffusion in a finite rectangle with time varying diffusion coefficient is established, that is, Eq. (12) and Eq. (13).

Stability and Convergence Analysis of ADI Model
Theorem 2: The truncation error of the ADI model (12)  Proof: According to the equations (10) and (11), simplify it and get Add the above two equations, we can get And subtract the above two equations, we can get Substitute (16) into (15), we can get ( 2 ) , 2 24 24 ( 2 ) , 2 24 24 24 24 From the equation (12), we can get that for any Theorem 4: The ADI model (12) of Chloride ions diffusion with time varying coefficients is convergent.
Proof: Combine with the Lax equivalence theorem [22], we can get the ADI model of Chloride ions diffusion is consistent, according to the definition of consistency. And base on it, stability is the necessary and sufficient condition for convergence. Hence, combine with the theorem 3, we can know the differential model is convergent.

Numerical Example
Now we consider Chloride diffusion in a rectangle reinforced concrete, with y L is 2900mm, x L is 100mm, the Chloride ion concentration of the surface of concrete 2 s C is 45% and 0 C is zero. Here take 1 h mm = and 1month τ = . Substitute these parameters to ADI model (12) and (13), then we can predict Chloride diffusion in concrete.
(1) Different α   Fig.3. 0 D is a fundamental diffusion coefficient, its increasing results in the acceleration of Chloride diffusion.  It is obvious that Chloride ion concentration will increase with time T . The larger time T is, the larger Chloride ion diffusion of is, and the worse the durability of concrete.
(4) Contour Distribution Fig. 5 shows the contour distribution of Chloride ion concentration in different situations, which describes Chloride diffusion in concrete more clearly. Figures (a) and (b)  , that is, about 25 years, Chloride ion will diffuse to everywhere in concrete, which is very dangerous in reality.

Conclusion
An ADI numerical model of two-dimensional Chloride ion diffusing problem in a finite rectangle with time varying diffusion coefficient is established, which is Eq. (12). ADI model (12)  Numerical example shows the effectiveness of ADI model, which can predict the diffusion of Chloride ion in concrete and reflect the influence of each parameter.