## 2 Materials and Methods
### Low Strain Integrity Testing  
Low strain impact integrity testing (also referred to as sonic, pulse-echo, PIT, or SIT) is a non-destructive test method that is widely used to determine the length and integrity of deep foundations. The method complies with ASTM D5882 [18] and other international codes. The elastic waves are produced by a hammer blow applied to the pile head surface and registered by an accelerometer attached to the pile head. The excited waves travel downwards along the pile body and reflect upward from the pile toe or non-uniformities such as cross-section changes, variations in elastic modulus, or density of the pile material. The recorded signals can be analyzed in both the time and frequency domains [40; 15; 41]. Data processing operations usually include integration, normalization, amplification, filtering, and averaging. Signals are converted from units of time (\(t\)) to units of length (\(L\)) according to the equation:  
\[L=\frac{V\cdot t}{2}, \tag{1}\]  
where \(V\) is the wave speed, assumed to be a constant value (typically 3600-4400 m/s for concrete piles). To verify the integrity of the pile and determine its length, the processed signals are analyzed along with site records, soil profile, and other relevant information. Signal features associated with reflections from a pile toe or structural defects are identified. A clearly pronounced reflection from a pile toe and a sufficiently stable signal are signs of a defect-free pile. When inspecting driven precast piles, special attention is to be paid to the splices between the sections. A splice can produce multiple reflection pulses, making it difficult to estimate the total length of a pile. As the resolution of the low strain method will vary depending on the frequency content of the input force pulses, it seems promising to use a set of hammers of different weights and head materials to access both the splice depth and the total length of a driven pile. Numerical Simulation of Low Strain Testing of Driven Precast Piles with Input Pulses of Different Durations  
To study the resolution of low strain integrity testing, particularly its ability to localize the splice and determine the total length of driven precast piles, full-wave numerical simulations were performed for four series of models. Each series consisted of 4 calculations with different durations of the input force pulse ranging from 0.5 to 2.0 ms. The numerical simulations were performed in the axisymmetric formulation using the finite element method and specialized software [14; 42], created in the COMSOL Multiphysics 5.3 package (Comsol Inc., Stockholm, Sweden). The calculations were carried out using a high-performance workstation based on two Intel Xeon processors (a total of 24 cores) and 512 GB RAM. To solve the system of equations, a direct solver MUMPS was used. The layouts of four synthetic pile models with three splice options are presented in Figure 2. Model 1 represents a single-section driven pile 14 m in length, and Models 2-4 represent piles of two jointed sections 8 and 6 m in length. Different types and conditions of pile splices were simulated with a 5 mm sand gap (Model 2), a 1 mm sand gap (Model 3), and a 1 mm air gap (Model 4) between the pile sections. The surrounding medium for the piles was modeled as dry sand. To avoid wave reflections from the external boundaries of the models, low-reflecting boundary conditions were imposed on the side and bottom boundaries of the surrounding soil. The physical properties of the materials specified in numerical models--density, pressure, shear wave speeds, and damping coefficients \(\alpha\) and \(\beta\) of the Rayleigh attenuation model [43; 44; 45; 46]--are given in Table 1.  
The modeling time duration was set to 10 ms. The time discretization was related to the spatial mesh size, and the time step was set to 1 us following the convergence condition