2. Grid generation

Generation of an adaptive mesh in the spatial domain is based on the r-refinement technique [14] which relocates a fixed number of nodal points to regions which need high spatial resolution in order to capture important characteristics in the solution. This has the benefit of improving computational effort in those regions of interest whilst using a fixed number of mesh points. The relocation of the fixed number of nodal points at any given time is achieved by solving Moving Mesh Partial Differential Equations (MMPDEs) [15, 16] derived from the Equidistribution Principle (EP). The EP [17] makes use of a measure of the solution error called a monitor function, denoted by M which is a positive definite and user defined function of the solution and/or its derivatives. Mesh points are then chosen by equally distributing the error in each subinterval. In this paper, MMPDE4 [15] is chosen to generate the adaptive mesh because of its ability to stabilise mesh trajectories and ability to give unique solutions for the mesh velocities with Dirichlet boundary conditions. MMPDE4 is given by

$$\frac{\partial}{\partial \xi} \left( M \frac{(\partial \dot{x})}{\partial \xi} \right) = -\frac{1}{\tau} \frac{\partial}{\partial \xi} \left( M \frac{\partial x}{\partial \xi} \right) \tag{2}$$

is used. It is composed of the standard arc-length monitor and the curvature monitor functions. Smoothing on the monitor function is done as described in [15]. Values of the smoothed

k¼i�p ð Þ Mk

where the parameter p is called the smoothing index which determines the extent of smoothing and is non-negative. γ is non-negative and is called the smoothing index and determines the

þ

un

x þ ununþ1 x þ unþ1 xx þ unþ1 xxxx � � ¼ un � δt

The variable spatial length of each interval is given by Hi where Hi ¼ Xiþ1ð Þ� t Xið Þt for

s ¼ x � Xið Þt

such that sEð Þ 0; 1 for every subinterval of the mesh (11). Define the septic Hermite basis

Consider the mesh on the domain ½ � a; b which is a solution of MMPDE4 given by

i ¼ 1, …, N. For some xE½ � Xið Þt ; Xiþ1ð Þt , define the local variable s as

unþ1 xx þ un xx 2 � �

þ

unþ1 xxxx þ un

x þ ununþ1 x � ð Þ uux n (9)

2 un

a ¼ X1ð Þt < X2ð Þt < … < XNþ1ðÞ¼ t b (11)

. Expression (9) is substituted into (1) and

xx þ un

Hið Þt (12)

xxxx � � (10)

2 � �

xxxx

http://dx.doi.org/10.5772/intechopen.71875

265

¼ 0 (8)

Piþp k¼i�p

Piþp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Numerical Simulation of Wave (Shock Profile) Propagation of the Kuramoto-Sivashinsky Equation Using an…

2 γ 1þγ � �j j k�i

vuuuut (7)

γ 1�γ � �j j k�i

monitor function M~ at the grid points are given by

rigidity of the grid.

3. Discretization in time

unþ1 � un δt � �

the terms are rearranged to give

unþ1 þ

functions with the local variables s as

The Crank-Nicolson scheme for the KSe is

Me ¼

þ ð Þ uux nþ1 þ ð Þ uux n 2 " #

uux

for the linearization of the non-linear term ð Þ uux nþ1

δt 2 unþ1 un

4. Septic Hermite collocation method

where δt is the time step. Rubin and Graves [18] suggested the expression

nþ1 ¼ unþ1

where τ is the relaxation parameter and it plays the role of driving the mesh towards equidistribution. Central finite difference approximation of MMPDE4 in space on the interval a ≤ x ≤ b gives

$$\frac{M\_{i+1} + M\_i}{2\left(\frac{1}{N}\right)^2} (\dot{\mathbf{x}}\_{i+1} - \dot{\mathbf{x}}\_i) - \frac{M\_i + M\_{i-1}}{2\left(\frac{1}{N}\right)^2} (\dot{\mathbf{x}}\_i - \dot{\mathbf{x}}\_{i-1}) = -\frac{E\_i}{\tau},\tag{3}$$

where

$$E\_i = \frac{M\_{i+1} + M\_i}{2\left(\frac{1}{N}\right)^2} (\mathbf{x}\_{i+1} - \mathbf{x}\_i) - \frac{M\_i + M\_{i-1}}{2\left(\frac{1}{N}\right)^2} (\mathbf{x}\_i - \mathbf{x}\_{i-1})\_i \qquad \text{i = 2, \dots, N} \tag{4}$$

$$\mathbf{x}\_1 = a \tag{5} \tag{5}$$

$$\mathbf{x}\_{N+1} = b. \tag{6}$$

The modified monitor function given by

$$M(\mathbf{x}, t) = \left(1 + \alpha^2 \left(\frac{\partial u}{\partial \mathbf{x}}\right)^2 + \alpha^2 \left(\frac{\partial^2 u}{\partial \mathbf{x}^2}\right)^2\right)^{\frac{1}{2}} \tag{6}$$

is used. It is composed of the standard arc-length monitor and the curvature monitor functions. Smoothing on the monitor function is done as described in [15]. Values of the smoothed monitor function M~ at the grid points are given by

$$\tilde{M} = \sqrt{\frac{\sum\_{k=i-p}^{i+p} (M\_k)^2 \left(\frac{\mathcal{V}}{1+\mathcal{V}}\right)^{|k-i|}}{\sum\_{k=i-p}^{i+p} \left(\frac{\mathcal{V}}{1-\mathcal{V}}\right)^{|k-i|}}} \tag{7}$$

where the parameter p is called the smoothing index which determines the extent of smoothing and is non-negative. γ is non-negative and is called the smoothing index and determines the rigidity of the grid.


*2.3. Measurement Techniques*

Electrochemical measurements were performed with an AUTOLAB/PGSTAT302N potentiostat (Version 4.9 AUTO83745, ECO CHEMIE, Utrech, Netherland, 2008) [34]. The central steel bar was used as the working electrode while the parallel bars on the sides were used as the counter electrodes. For the electrochemical measurements a silver/silver chloride reference electrode was used (SSCE; +0.222 V SHE). During the measurements a damp flannel was placed on the mortar surface in order to improve electrolytic contact between the mortar and reference electrode (Figure 2).

**Figure 2.** Setup for carrying out electrochemical measurements using an AUTOLAB/PGSTAT302N potentiostat and setup of the accelerated corrosion process using a POWER SUPPLY FAC-662B POWER SUPPLY FAC-662B (PROMAX, Barcelona, Spain, 1998).

All the electrochemical measurements were associated with the humidity content of the test specimens, which were gradually moistening and were kept in a humidity chamber.

The use of the potentiostat enabled the evolution of the corrosion to be studied by means of the polarization curve, producing data on the open circuit corrosion potential (Ecorr), and through the polarization resistance on the corrosion rate (icorr). Since midway through the 20th century the polarization resistance technique has become a common technique to study corrosion [35].

The first evaluation of rebar behavior with regard to corrosion was made by measuring their corrosion potentials, which are useful for a qualitative determination of whether the steel bars are in an active or a passive state. The data obtained were interpreted according to the ASTM C 876 standard [36], from which, taking into account the reference electrode used, it was concluded that Ecorr < −231 mV would indicate a 90% probability that corrosion exists in the active state, a potential in the region −231 mV < Ecorr < −91 mV would indicate uncertainty, and Ecorr > −91 mV would indicate a probability of 10% that corrosion exists in the active state.

The following step was to interpret the results for the rate or intensity of corrosion as this information enables us to quantitatively interpret the active or passive state of the bars.

According to different research [37], icorr < 0.1 μA/cm2 indicates a passive state, 0.1 μA/cm2 < icorr < 0.5 μA/cm2 is equivalent to a low level of corrosion, 0.5 μA/cm2 < icorr < 1 μA/cm2 to a high level of corrosion, and icorr > 1 μA/cm2 to a very high level of corrosion.

The central bar was connected to a POWER SUPPLY FAC-662B (PROMAX, Barcelona, Spain, 1998) to accelerate the corrosion process in both the standard test specimens and the test specimens with LFS, for specimens containing 1.2% and 2% of mixed-in chloride ions by weight of cement, making the embedded central steel bar act as an anode. On the upper face of the test specimens was placed a damp flannel with a lead sheet on top; this acted as a continuous counter electrode [38]. The voltage necessary to maintain the intensity of the preset current (1.3 mA) was then measured, moistening the test specimens frequently, as the potential varies substantially with the humidity content (Figure 2).

#### **3. Results and Analysis**

Figure 3a shows the evolution of the corrosion potential (Ecorr) in standard test specimens while Figure 3b provides the data from test specimens with slag for different chloride ion percentages. The corrosion potentials in the standard test specimens for chloride ion percentages equal to or below 0.4% are clearly less negative than those in the test specimens with chloride ion percentages above the limit set by the standards of different countries, which show potentials suggesting the active state. In the test specimens with LFS, regardless of the chloride ion percentages possessed by the test specimens, the corrosion potentials correspond to the active state. In all test specimens the potential becomes more negative over time and, consequently, with their humidity content, as well as with the chloride ion percentages of the test specimens.

**Figure 3.** Evolution of Ecorr over time in standard test specimens (**a**) and in LFS specimens (**b**) at different chloride ion percentages.

The corrosion rate increases over time and, therefore, with the humidity content, together with the chloride ion percentage introduced at the time of kneading. In the test specimens with chloride ion percentages within the limit of the EHE "Instrucción de Hormigón Estructural" Instruction (Figure 4a) (0.4% of chloride ions by weight of cement), the corrosion rates are characteristic of the passive state as the maximum values are around 0.1 μA/cm2, regardless of whether the test specimens contain LFS, with data corresponding to 545 days of exposure in the humidity chamber. In the test specimens with chloride ion percentages above 0.4% by weight of cement (Figure 4b), the corrosion rates are characteristic of the active state, with values above 1 μA/cm2. In the test specimens with 0.8% chloride ions, with data obtained for up to 545 days of exposure, the corrosion rates are slightly higher in the standard test specimens than in the test specimens with LFS. In the test specimens with chloride ion percentages of 1.2% and 2%, data are held on the corrosion rate for up to 326 days of exposure in the moist chamber, as from that moment they underwent an accelerated corrosion process; the results are discussed below. In these test specimens, the corrosion rates increased with the chloride percentage contained in the test specimens, which were slightly higher in the standard test specimens than in the LFS test specimens.

**Figure 4.** Evolution of icorr over time in test specimens without chlorides and with 0.4% chloride ions by weight of cement (**a**) and in test specimens with chloride ion percentages of 0.8%, 1.2% and 2% (**b**).

Figure 5 shows the evolution of the voltage needed to maintain the preset current at a level of 1.3 mA. The voltage increases over time and with the chloride ion percentage introduced at the time of kneading. Moreover, standard specimens need higher voltage to maintain the preset current than do the specimens with slag LFS.

Figures 6 and 7 show the symptoms produced by the corrosion of the steel bars subjected to an accelerated corrosion process by an impressed anodic current for chloride ion percentages of 1.2% and 2.0% by weight of cement. Each image shows the number of days that had elapsed since the start of the process of natural corrosion, the number of days subjected to accelerated corrosion, and the potential and the intensity of the current passed through the rebar.

**Figure 5.** Evolution of the potential (V) necessary to maintain a constant current of 1.3 mA in each rebar.

**(b)**

**Figure 6.** Symptoms in the MCC test specimen with 1.2% chloride ions (**a**) and 2% chloride ions (**b**).

**(b)**

**Figure 7.** Symptoms in the MCE test specimen with 1.2% chloride ions (**a**) and 2% chloride ions (**b**).

In the standard test specimen with a chloride ion percentage of 1.2%, it was observed that the first rust stains began to appear 31 days after the start of the accelerated corrosion, appearing both on the edge of the test specimen and on the lower face, gradually increasing up to 72 days when the stain extended over the entire face of the test specimen, while on the upper face there were no signs of corrosion. The standard test specimen showed no signs of cracking after the central bar had been connected to the current for 282 days, an externally applied electrical charge of 366.6 mA having passed through the central bar (Figure 6a).

In the final test specimen which was subjected to the accelerated corrosion process, corresponding to the other standard test specimen with 2% chloride ions, rust stains began to appear on the edge of the test specimen 11 days after the start of the process, with stains appearing on the lower face of the test specimen after 32 days. From that moment, the rust stains became more uniform on the lower face of the test specimen after 73 days with an electrical charge passed through of 94.9 mA. The test specimen continued to show signs of corrosion over time in the form of increasingly marked stains, but it presented no cracks after 283 days of accelerated corrosion with an electrical charge of 367.9 mA having passed through the bar (Figure 6b).

In the test specimens with LFS and with 1.2% chloride ions by weight of cement, rust stains began to appear in the upper part of the test specimen coinciding with the position of the bar which was undergoing accelerated corrosion. This initial stain appeared approximately 40 days following the commencement of the accelerated corrosion process and continued to enlarge its surface area and importance with the progressive passage of the current. After 174 days of current, the first cracks appeared, on both the upper and lower faces of the test specimen and even on its edge, coinciding with the central bar. The cracks increased in length and width over time, forking at the end of the test specimen 250 days after the commencement of accelerated corrosion, an overall electrical charge of 325 mA having passed through the bar. The crack width after 282 days, once a charge of 366.6 mA had passed through the bar, was 1 mm (Figure 7a).

The first rust stain in the test specimen with 2% chloride ions and with LFS appeared 42 days after the commencement of accelerated corrosion. The first stain, on the upper face of the test specimen, increased in size over time, leaving an appearance of generalized stains on the test specimen. After 132 days since the commencement of the accelerated corrosion process, the first cracks appeared, coinciding with the central bar and appearing both on the upper and lower faces of the test specimen. The width of the cracks gradually increased; after 250 days and after passing a 325 mA electrical charge through the central bar, numerous cracks in a mesh were formed on the rear of the test specimen, which caused the mortar to disintegrate (Figure 7b).