## 3 Simulation and Analyses  
### Two-Dimensional Rough Surface Model Setup  
Finite element simulation is an effective method for studying ultrasonic detection, which involves multi-physics coupled field. A two-dimensional rough surface model is developed with commercial software (COMSOL Multiphysics 5.2a), as shown in Figure 4a. It is used to analyze the relationship between the surface roughness parameters (\(\mathbf{R_{a}}\) and \(\mathbf{R_{sm}}\)) and time-domain signal of micro-crack. The model consists of a virtual focused transducer, coupled water, and a 9 mm thick steel plate with a back-surface micro-crack. The depth of the micro-crack is normally less than five percent of the wall thickness of the workpiece. The focused transducer, formed by a piezoelectric wafer and arc lens, has chord length of 8 mm and curvature radius 25 mm. It is at a stand-off distance of 20 mm from the front surface of the steel plate. The material properties used in the model are obtained from the material library of the software. The longitude wave velocity in water is 1480 m/s and shear wave velocity in steel is 3230 m/s. The excitation signal applied to the external circuit is a broadband modulated pulse, which can be modeled as a Gabor function [35]:  
\[\mathbf{f}(t)=A\cdot\exp(-\frac{(t/2-\mu)^{2}}{\sigma^{2}})\cdot\sin(2\pi f_{0} t) \tag{4}\]  
where \(A\) is the reference amplitude and center frequency \(f_{0}\) of the transducer is set to 5 MHz. \(\mu=1/f_{0}\) and \(\sigma=1/(2f_{0})\) correspond to the translation and standard deviation of Gaussian function.  
Figure 4: (**a**) The two-dimensional simulation model; (**b**) partial view of rough sinusoidal surface.  
The liquid-solid interfaces, including the front surface and back surface and the arc lens surface in the model, are set to be acoustic-structure coupling boundaries. Other boundaries are set to be radiation boundaries in the water region and low reflection boundaries in the steel region. The grid control region is used to mesh the refined grid to improve calculation accuracy. The grid density is set to 8 elements per wavelength in the control region and 6 elements per wavelength in the other region. The simulation study is carried out with the transient analysis module. The temporal resolution is set to 20 time steps per period of the acoustic wave. The time domain signals are captured using the sampling time interval \(\Delta t\) = 0.01 \(\upmu\)s. This study is on the rough surface, and a surface with sinusoidal curves is used to model the rough surface [7; 28]. Amplitude and wavelength of the curves are used to describe the arithmetic average heights and the horizontal parameter. Figure 4b shows a partial view of the sinusoidal surface with different \(\mathbf{R_{a}}\) and \(\mathbf{R_{sm}}\). The mathematical expression corresponding to the roughness parameters in the model can be described as:  
\[x=range(0,\Delta x,x_{1}) \tag{5}\]  
\[y=B\cdot\sin\{range(0,0.02\times\pi,m\times\pi)\} \tag{6}\]  
where \(x_{1}\) represents the end of the rough interface and \(\Delta x\) corresponds to \(\mathbf{R_{sm}}\). Half of the amplitude \(B\) of the sinusoid is chosen to be of similar extent to \(\mathbf{R_{a}}\). In addition, \(W\) represents the width of the crack and \(D\) represents the crack depth.