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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.

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 [

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 [

Recently, many meshless methods have been proposed in literature [

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

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

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

The cubic spline weight function is defined by

The distribution of chloride in the problem domain

The essential concept of MWLS is that the method is a weighted residual method; that is, the weight function is residual and the function

The residual of (

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

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,

The analytical solution for the 1D diffusion of chloride ions in concrete is

RMS (%) error of different weight functions at 31 nodes for exposure time

| Quartic spline | Cubic spline | Gaussian function ( | |||
---|---|---|---|---|---|---|

MWLS | EFG | MWLS | EFG | MWLS | EFG | |

| 0.1888 | 0.1300 | 0.1441 | 0.0622 | 0.1725 | 0.8256 |

| 0.1485 | 0.1626 | 0.1097 | 0.0877 | 0.1397 | 0.8219 |

| 0.1068 | 0.2518 | 0.1008 | 0.1854 | 0.1211 | 0.8944 |

Table

Results of MWLS, EFG, and FEM and exact solutions at a few specific locations (31 nodes,

Location (m) | Chloride concentration (%) | |||
---|---|---|---|---|

| MWLS | EFG | FEM | Exact |

0.01 | 4.1125 | 4.1132 | 4.1174 | 4.1129 |

0.03 | 2.5014 | 2.5067 | 2.5110 | 2.5058 |

0.05 | 1.3109 | 1.3134 | 1.3096 | 1.3112 |

0.07 | 0.5889 | 0.5855 | 0.5800 | 0.5826 |

0.09 | 0.2268 | 0.2207 | 0.2182 | 0.2180 |

0.11 | 0.0754 | 0.0702 | 0.0690 | 0.0682 |

0.13 | 0.0229 | 0.0196 | 0.0186 | 0.0178 |

0.15 | 0.011 | 0.00087 | 0.007637 | 0.0039 |

Figure

Variation of RMS error at different nodes.

Variation of computational time with number of nodes.

Figure

Variation of chloride concentration with depth.

In the second example, when the diffusion coefficient

The results shown in Figure

The initiation period of corrosion by different methods (at 31 nodes).

Initiation period (year) | MWLS | EFG | FEM | Exact solutions |
---|---|---|---|---|

| 4.50 | 4.55 | 4.60 | 4.67 |

| 6.00 | 6.35 | 6.40 | — |

Chloride concentration-depth at time = 20 years.

From these two examples, the MWLS and EFG methods appear to be effective in accurately predicting chloride concentrations in concrete structures. The accuracy of the EFG is higher than that of MWLS, but the error of MWLS remains small in actual engineering applications.

The rate of chloride diffusion in an actual reinforced concrete structure is very slow, and thus measuring long term diffusion is a slow, time-consuming process. In addition, few research studies have dealt with chloride diffusion over long periods. In this section, a concrete plate of 0.15 m × 0.15 m was used as an example, as shown as Figure

2D chloride diffusion in a square concrete plate.

A regular distribution of

Table

Comparison of MWLS results with EFG, FEM, exact solutions at the specified locations (

Location | Chloride concentration (%), exposure time | ||||
---|---|---|---|---|---|

MWLS | EFG | FEM | Exact solutions | ||

| | 0.4190 | 0.3926 | 0.3909 | 0.3909 |

| 0.2894 | 0.2363 | 0.2358 | 0.2355 | |

| 0.2008 | 0.1501 | 0.1501 | 0.1497 | |

| 0.1455 | 0.1151 | 0.1151 | 0.1144 |

In the EFG method,

Comparison of results with different

Location | Chloride concentration (%), exposure time | ||||||
---|---|---|---|---|---|---|---|

| |||||||

| | | | | Analytical solution ( | ||

| | 0.3926 | 0.3921 | 0.3922 | 0.3936 | 0.4157 | 0.3909 |

| 0.2363 | 0.2361 | 0.2361 | 0.2360 | 0.2402 | 0.2355 | |

| 0.1501 | 0.1520 | 0.1500 | 0.1498 | 0.1476 | 0.1497 | |

| 0.1151 | 0.1150 | 0.1150 | 0.1148 | 0.1108 | 0.1144 |

In the following study the chloride diffusion coefficient

The initiation period of corrosion as calculated by different methods.

Initiation period (year) | MWLS | EFG | FEM | Exact solutions |
---|---|---|---|---|

| 3.00 | 3.15 | 3.30 | 3.25 |

| 3.30 | 5.60 | 5.85 | — |

Distribution of chloride concentration: (a) MWLS, (b) EFG, and (c) FEM at exposure time

MWLS

EFG

FEM

It can be seen that the initiation period of corrosion computed with MWLS is different from the results given by EFG and FEM when

An eight-year observation of chloride penetration in a real marine structure has been reported in [^{2}/s and

Comparison of chloride concentration computed by MWLS and EFG and experimentally measured data after exposure time: (a)

These results still indicate the existence of an error between the numerical simulations and the experimental measurements. In the actual situation, the diffusion coefficient

In this reported study, a MWLS method based on the weighted least squares approach and EFG method were used to address the problems in the simulation of chloride diffusion in concrete. A discrete function was adopted in the MWLS method, which avoided tedious numerical integration. A variety of 1D and 2D numerical examples demonstrated that the accuracy of MWLS is close to EFG, FEM, and an analytical solution. The EFG method provided an initiation period of corrosion in concrete that was close to FEM. However, the MWLS consumed much less computation time than EFG. Hence, the MWLS is an acceptable meshless method when computation time is considered, whereas the EFG is suitable when accuracy is considered. Both the MWLS and EFG methods successfully predicted the chloride concentration in concrete structure, which is used to prevent reinforcement corrosion in concrete structures.

The authors declare that they have no competing interests.