## 3 Establishing Finite Element Models  
Layered pipeline structures are susceptible to corrosion and stress damage, due to their direct contact with, and exposure to, the transported material during material transport. The amount of damage, and the circumferential and axial positions of damage that occurs during use are complex. One typical and urgent issue to be addressed is the difficulty in determining the axial position of defects when double defects occur in layered pipeline structures during engineering applications, and the distance between the defects is relatively close. This is because the wave packets of the defect reflection waves overlap, making it challenging to identify the defect reflection wave packets, to determine the axial position of the defects. To address the above-mentioned issue, this study uses the finite element simulation software ABAQUS to establish a model of a layered pipeline structure. The study analyzes the sensor signals under different damage quantities and circumferential positions, calculates the damage index values, and constructs two-dimensional damage index matrices based on the relative axial position of the damage and the damage index value. Additionally, a three-dimensional damage index matrix is constructed, based on the relative axial position of the damage, the quantity and angle of the damage, and the damage index value. These matrices are used to locate the axial position of defects in layered pipeline structures, based on the damage index values.  
The pipeline model established by finite element simulation is a layered pipeline structure composed of a steel pipe structural layer (inner layer), a rigid polyurethane foam insulation layer (middle layer), and a high-density polyethylene anticorrosive layer (outer layer). The material properties of each layer are shown in Table 1. The actuator is composed of 16 piezoelectric elements, which are centrally coupled at a distance of 30 mm from the near end of the steel pipe structural layer. The sensor is composed of four piezoelectric elements, which are centrally coupled at a distance of 600 mm from the near end of the steel pipe structural layer [27]. This arrangement is designed to suppress the generation of other modes [28], and to ensure that the reflected waves from the damage and the reflected waves from the end are separated. All the research in this paper is based on a layered pipeline structure with different relative axial positions of double defects. Lead zirconate titanate (referred to as PZT-4) was selected as the piezoelectric element in the test system. For the sake of brevity and clarity in the text, the layered pipeline structure is referred to as a pipeline structure; the layered pipeline structure with double defects at the same circumferential position but different axial positions is referred to as a pipeline structure with double defects at the same circumferential position; and the layered pipeline structure with double defects at different circumferential and axial positions is referred to as a pipeline structure with double defects at different circumferential positions.  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline
**Material Properties** & **Internal Diameter/mm** & **Outer Diam-** & **Wall Thick-** & **Den-** & **Modulus of** & **Poisson** & **Tube** \\ \hline Steel tube & 68 & 76 & 4 & 7850 & \(2.1\times 10^{11}\) & 0.32 & 2.7 \\ Rigid polyurethane foam & 76 & 136 & 20 & 80 & \(7.8\times 10^{8}\) & 0.25 & 2.1 \\ High-density polyethylene & 136 & 140 & 2 & 946 & \(5.52\times 10^{8}\) & 0.4 & 2.1 \\ \hline \end{tabular}
\end{table}
Table 1: The pipeline structure material properties.