## 3 Data Collection
### FEM Modeling  
The explicit finite element method was used for the numerical modeling of ultrasonic wave propagation through asphalt concrete, as it is a direct integration method using the principle of central differential and a diagonal matrix mass, which is also utilized in this paper for wave propagation simulation, as recommended [6].  
The effects of crack size and location on wave propagation were analyzed. Such analysis can be simulated since ultrasonic wave propagation has an inverse relationship with a medium containing voids or cracks. With the increase of voidage, the wave velocity propagated through the medium decreases, which is due to the fact that the medium had large gaps containing air and wave propagation through air is slower than that in aggregates. Also, when ultrasonic waves propagate through materials with large voids, sound energy attenuates faster, which causes more dispersion of longitudinal waves. Due to this dispersion, many equivalent scatters of longitudinal wave are formed, which greatly affects ultrasonic wave propagation, resulting in a long travel time of the ultrasonic wave.  
A two-dimensional model of space size with dimensions of 300 mm \(\times\) 150 mm was constructed. Loading surf-1 on top of model with a length of 40 mm in the assembly module was created for ultrasonic wave input, which was loaded under sinusoidal stress wave with a load magnitude of 0.1 MPa, as suggested [13]. The amplitude curve was defined to describe the change in load and boundary conditions with time by the Fourier series expression, as shown in Equations (3) and (4).  
\[a=A_{\circ}+\sum_{n=1}^{N}[A_{n}\cos n\omega(t-t_{\circ})+B_{n}\sin n\omega(t- t_{\circ})](t\geq t_{\circ}), \tag{3}\]  
\[a=A_{\circ}(t<t_{\circ}), \tag{4}\]  
where N represents the number of Fourier series in the formula, \(\omega\) represents the circular frequency (rad/s), \(t_{\circ}\) is the initial time, \(A_{\circ}\) is the initial amplitude, \(A_{n}\) and \(B_{n}\) are the coefficients, and n = 1, 2, 3, 4,..., N.  
Figure 2a shows a schematic representation of ultrasonic wave propagation generated by an ultrasonic transducer through pavement without cracks. The bottom asphalt layer was selected as the analyzing point. Figure 2b shows that three cracks, A, B, and C, of varying pavement depths, were created. In the first-round simulation, only cracks A and B were generated in order for the effect of crack depth on wave propagation to be checked. In the second-round simulation, only crack C was set and its two crack widths were utilized to evaluate the effect of crack size.  
The simulation frequency was selected as 50 kHz, which equals the frequency that was applied in the field NDT ultrasonic tests. The frequency can be realized in finite element simulation software ABABQUS by defining amplitude curve with circular frequency \(\omega=2\pi\)f = 314,159 rad/s, with initial time and initial amplitude equal to 0.  
The selection of a finite element number per wavelength is critical and greatly affects simulative accuracy. At least seven finite elements per wavelength is suggested to achieve satisfyingly accurate results [14; 15; 16]. In this study, the element per wavelength was determined as 20 based on a sensitivity analysis, which was conducted to determine the proper element number per wavelength. This ensured the accuracy of the solution and did not significantly decrease the calculation efficiency at the same time. It should be noted that whenever the FEM was varied (i.e., mix type, pavement depth, or crack location), sensitivity analysis needed to be re-conducted, since the element number per wavelength was highly sensitive to material property and crack size. Such a step is necessary to ensure simulation efficiency and accuracy. As an example, for the AC-13 mix without cracks, the element number was determined as 20. Taking the asphalt mixture as homogenous, the free meshing technique was adopted and the CPS4R planar quadrilateral stress unit was selected as the cell type.