Advances in Materials Science and Engineering

Volume 2016 (2016), Article ID 3824835, 10 pages

http://dx.doi.org/10.1155/2016/3824835

## Prediction of Chloride Diffusion in Concrete Structure Using Meshless Methods

^{1}State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China^{2}School of Human Settlements and Civil Engineering, Xi’an Jiaotong University, Xi’an 710049, China

Received 29 June 2016; Revised 10 October 2016; Accepted 16 November 2016

Academic Editor: Luciano Lamberti

Copyright © 2016 Ling Yao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Degradation of RC structures due to chloride penetration followed by reinforcement corrosion is a serious problem in civil engineering. The numerical simulation methods at present mainly involve finite element methods (FEM), which are based on mesh generation. In this study, element-free Galerkin (EFG) and meshless weighted least squares (MWLS) methods are used to solve the problem of simulation of chloride diffusion in concrete. The range of a scaling parameter is presented using numerical examples based on meshless methods. One- and two-dimensional numerical examples validated the effectiveness and accuracy of the two meshless methods by comparing results obtained by MWLS with results computed by EFG and FEM and results calculated by an analytical method. A good agreement is obtained among MWLS and EFG numerical simulations and the experimental data obtained from an existing marine concrete structure. These results indicate that MWLS and EFG are reliable meshless methods that can be used for the prediction of chloride ingress in concrete structures.

#### 1. Introduction

Reinforced concrete (RC) structures form the basis for most construction in civil engineering. However, a considerable number of reinforced concrete structures cannot achieve its design service life because of premature durability problems. Many factors influence the durability of a structure, including chloride ingress, carbonation resulting from penetrating carbon dioxide, and moisture transport. Extensive research has shown that chloride ingress in concrete is one of the most significant processes that can seriously impair the long term durability of RC structures [1–3]. Conventional testing of chloride penetration into concrete is time-consuming, making it advantageous to numerically simulate chloride ingress into concrete.

Many studies have focused on Fick’s second law of diffusion as the basis for the description of chloride transport in concrete, assuming that diffusion is the dominant transport mechanism. However, obtaining a sound analytical solution can be difficult in practical engineering of complicated structures. Therefore, development of more effective methods for predicting chloride concentration in concrete structures is necessary. Many researchers have proposed numerical simulation methods to describe the phenomenon of chloride transport in concrete [4–6]. In general, these methods are based on the finite element methods (FEM), the finite difference methods (FDM), or similar approaches. However, these methods are limited in terms of overcoming problems, such as complex pretreatment and mesh generation. In contrast, meshless methods only employ data at the nodes and hence do not require that the solution domain be subdivided into many smaller regions. Consequently, meshless methods have a simple preprocessing step and offer high accuracy for scientific and engineering problems.

Recently, many meshless methods have been proposed in literature [7–14], including the smoothed particle hydrodynamic method, the diffuse element method, the EFG method, the reproducing kernel particle method, the finite point method, and the meshless local Petrov-Galerkin method as well as many others. The point collocation methods and the Galerkin method of discretization are among the most commonly used meshless methods. The Galerkin method is a famous example of a meshless method, which was proposed by Belytschko et al. in 1994 [7]. The EFG method is a type of Galerkin meshless method. The other type of a widely used meshless method is the point collocation method. The meshless weighted least squares (MWLS) method is a type of point collocation methods. The MWLS and EFG methods have been successfully applied to solve problems regarding conduction of heat transfer [15]. However, few studies have reported on the application of meshless methods to solve the problems of chloride transport in concrete. In reality a variety of transport mechanisms occur, but chloride transport in concrete is modelled as a diffusion process in current research. Bitaraf and Mohammadi in [15] reported a FPM developed and adopted for solving the chloride diffusion equation in concrete for prediction of service life of concrete structures and initiation time of corrosion of reinforcements. Guo et al. in [16] reported the use of the transient meshless boundary element method for predicting chloride diffusion, which emphasized time-dependent nonlinear coefficients.

In the study reported in this paper, the MWLS and EFG methods were used to solve problems of chloride transport by diffusion in concrete structure. This paper is organized as follows. Section 2 presents a brief introduction of the moving least squares (MLS) approximations. The implementation of EFG and MWLS methods is shown in Sections 3 and 4. The numerical examples are demonstrated and the results compared with other methods in Section 5. Finally, Section 6 presents the concluding remarks.

#### 2. MLS Approximation Scheme

MLS approximation is a well-known meshless interpolation scheme. MLS is adopted as an approximation scheme in MWLS and EFG methods. In MLS approximation, the function is approximated by as follows:where is a complete polynomial basis of order, is a vector containing coefficients, , which are functions of the coordinates . For a linear basis , and whereas for a quadratic basis . is a function of the weighted residual, which represents the approximated values of the field function at the nodes, , and, which is a weight function, and is the nodal parameter of the field variable at node in which the coefficient can be chosen to minimize the weighted residual. Substituting (5) into (1) yieldswhere is a shape function.

In the MLS approximation, the continuity relates not only to the basic function but also to the weight function. The weight function plays various important roles, the first of which is to provide weighting of the residuals at different nodes in the support domain. The second role is to ensure that the nodes leave or enter the support domain in a gradual (smooth) manner when moves, thereby ensuring the compatibility condition [17]. In this study, the weight function is selected as follows.

The cubic spline weight function is defined bythe quartic spline weight function is defined byand the normalized Gaussian weight function is given throughwhereand represents the dimensionless scaling parameter and , are the distances to the nearest neighbors at node.

#### 3. EFG Method for Chloride Diffusion in Concrete

The distribution of chloride in the problem domain is governed by Fick’s second law:where represents concentration, is time, and , represent the diffusion coefficients. In this article, assuming is the same in and directions, the initial concentration of chloride ion present in concrete isDirichlet’s boundary condition is given by The weak form of equation (13) is expressed asThe function can be written asLet , and thenwhere isLagrange’s Multiplier technique was used to impose the essential boundary conditions in the EFG method; hence, substituting (20), (21), and (22) into (16) we obtainFinally, (24) can be written aswhereThe time interval [] is subdivided into a finite number of equal subintervals . By using the *θ* method, (25) can be written asEquation (25) can be written aswithwhere , , and are the Crank-Nicolson form, Galerkin form, and backward difference form, respectively.

#### 4. MWLS Method for Chloride Diffusion in Concrete

The essential concept of MWLS is that the method is a weighted residual method; that is, the weight function is residual and the function is obtained by summation of the squares of residuals., are the weight coefficients, which when minimized, producewhere . are the test functions. A discrete form is adopted to avoid integration as follows:where , are the weighted coefficients, is the collection node, and and are the number of evaluation points needed to satisfy the governing equation and the boundary conditions, respectively. An approximation function of residuals is set with the MLS scheme. In the present study, the penalty function approach was used to impose the essential boundary conditions in MWLS.

The residual of (31) is set according to (32):where [18], is the penalty parameter required to apply the boundary condition, and is the characteristic length in this problem. By adopting the variational principle in (31) and using the discrete form to avoid integration, finally this results intoThe above equations are the computation format of MWLS with respect to chloride diffusion in concrete. Equation (34) is solved by the approach in the EFG method.

#### 5. Numerical Examples

In the previous sections, the efficiency of the methods has been verified by using one-dimensional (1D) and two-dimensional (2D) numerical examples to demonstrate the applicability of the proposed method for quantifying chloride ion diffusion in concrete structures. In the current analysis, one Gauss point in the 1D problem and Gauss points in the 2D problem were used to perform integration in the EFG method. MATLAB codes were developed to obtain the EFG and MWLS results, whereas the FEM results were obtained using the same four-noded brick elements in COMSOL Multiphysics 4.3 software.

##### 5.1. 1D Case Study

The first example is that of a concrete slab of 0.15 m thickness. The left boundary is permanently subjected to a constant chloride concentration of 5% (by mass of NaCl). The initial chloride concentration is 0, , and for a linear basis . In the first example, the diffusion coefficient is assumed to be constant and in the second example to be a time-dependent function , where is the diffusion coefficient at some reference time and denotes a material constant [18]. For this situation, have been chosen. For the purpose of convergence studies, the root-mean square (RMS) error is defined as , where is the number of sample points, is the calculation result with simulation, and denotes the result obtained with the analytical solution.

###### 5.1.1. Coefficient Is a Constant

The analytical solution for the 1D diffusion of chloride ions in concrete iswhere is the error function and is assumed. The slab is discretized into 31 nodes with an exposure time years to implement the meshless methods. Table 1 shows the numerical comparison of the RMS error for different weight functions and *θ* with MWLS and EFG methods. The results indicate that the observed minimum error occurs when the cubic spline weight function is used and using the MWLS method. Therefore, the cubic spline weight function was chosen with in the numerical simulation that follows with the MWLS method. When using the EFG method, the minimum error occurred when the cubic spline weight function and were adopted, and hence the cubic spline weight function and were used in the EFG method analysis.