## 1 Introduction  
Many factors contribute to the deterioration of civil infrastructures [1; 2; 3], and among them, corrosion exists in almost all structures [4; 5; 6] and is often responsible for various structural damages [7; 8; 9], such as cracks, reduced stiffness, and eventually reduced or lost bearing capacity. Prestressed concrete structures have been widely used in modern concrete infrastructures because of its advantage of high stiffness, crack resistance, and good durability [10], and the post-tensioned method is often used in the production of prestressed concrete structures [11]. However, insufficient grouting of the prestressed tension duct provides channels for water/moisture migration and chloride diffusion [12], resulting in the corrosion of prestressing tendons, which may lead to the severe degradation of the bearing capacity of post-tensioned prestressed concrete structures [13; 14; 15]. Therefore, an effective technique to evaluate the grouting compactness should be developed. Various non-destructive testing (NDT) techniques, including the impact-echo (IE) method [16; 17], the X-ray penetration method [18], the ultrasonic testing method [19; 20; 21; 22], and the ground penetrating radar (GPR)-based method [23] have been developed to detect the grouting compactness. However, the above methods are usually performed with a structural downtime, which make these methods unsuitable for the real-time monitoring of civil structures. With the advantage of real-time capacity, structural health monitoring (SHM) has become an emerging topic in the field of civil engineering [24; 25; 26; 27]. Lead Zirconate Titanate (PZT) transducers have been widely studied and applied in the field of SHM due to their multiple advantages of wide bandwidth [28; 29], quick response [30; 31], dual capacity as a sensor and an actuator [32], and energy harvesting [33; 34; 35]. In addition, various packaging and protection methods have been developed for PZT transducers [36; 37; 38] so that they can be easily deployed for real-time structural monitoring through embedment for concrete structures [39; 40; 41] or surface bonding for steel structures, concrete structures, and composite structures [42; 43]. PZT-based structural health monitoring methods include the active sensing approach [44; 45; 46; 47], electro-mechanical impedance (EMI) approach [48; 49; 50], and passive sensing method [51; 52; 53]. Among these methods, the PZT-based active sensing method has been used to monitor the grouting compactness. [PERSON] et al. [54] proposed a detection approach based on the time reversal theory to detect the compactness of concrete, in which two damage indexes were defined to determine the existence and location of artificial and real damage areas. [PERSON] et al. [55; 56] used the PZT-based active sensing method to monitor the grouting compactness in a post-tensioning tendon duct (PTTD) and established a two-dimensional finite element model (FEM) to further explain the wave propagation in a PTTD with different grouting compactness. [PERSON] et al. [57] developed a new PZT-enabled time-reversal method by installing a PZT ring on the prestressed steel to monitor the grouting quality of a post-tensioning tendon duct in real-time. An ultrasonic time difference of arrival (TDOA) algorithm was applied by [PERSON] et al. [58] to monitor the grouting compactness of the concrete-filled fiber-reinforced polymer tube (CFFT). However, it limits the use of the PZT-based active sensing approach in that it requires two or more sensors to generate and detect the stress wave. The EMI is another powerful technique for a PZT-based SHM. In this technique, a PZT patch, which is bonded on or embedded into the structure, acts as both a sensor and an actuator [59]. The basic principle of the EMI technique is to track the electrical impedance of the PZT patch bonded on or embedded into the structure.