# Determination of Concrete Formwork Removal Time Based on Ultrasound Reflection
## 5 Numerical Simulations
The reason is that higher frequencies correspond to shorter periods and thus sharper signals, allowing easier differentiation of possible reflections and different wave modes, as evidenced in the snapshots of Figure 8, and later in the waveforms. Figure 8a shows the displacement field approximately 20 us after the excitation for the case of fresh concrete (\"concrete 1\" in Table 1). The longitudinal wavefront propagates in the depth of the buffer towards concrete as well as on the surface reaching the receiver, while the Rayleigh wave follows. Figure 8b, shows the field after approximately 30 us. Now part of the wave has already been transmitted into the concrete while the reflection from the interface is also visible. In Figure 8c the reflection is close to the receiver. Figure 9 shows the corresponding displacement fields for concrete hardening stage #3. The point of interest is that the reflection is much weaker, and it is essentially negligible in Figure 9c when it is supposed to reach the receiver. Finally, for concrete #10, corresponding to hardened conditions according to Table 1, the reflection becomes, again, visible and even stronger (see Figure 10b,c). Indicative waveforms as recorded by the sensor for three different stiffness values of concrete can be seen in Figure 11a. One corresponds to fresh concrete (#1 in Table 1), the second to an intermediate stiffness (#3) and the last to fully hardened concrete (#10). Correspondingly to the experimental waveforms, all three numerical waveforms share exactly the same initial part (until about 40 us) because of the longitudinal wave travelling on the surface (visible at 15 us) and the strong Rayleigh portion between 27 us and 40 us. The Rayleigh part does not resemble the single cycle of the excitation because of the \"aperture effect\" [25]. The Rayleigh wavelength for 1 MHz is less than 2 mm, much smaller than the sensor size of 15 mm. Therefore, while the whole Rayleigh portion is within the receiver boundaries, positive and negative phases are essentially eliminated allowing indications only when it enters (approximately at 27 us) and when it exits (40 us) the physical size of the sensor. The differentiation between the three waveforms comes after 40 us, when the longitudinal reflection from the buffer arrives. For the case #1 (red line, \begin{table}
\begin{tabular}{c c c c} \hline
**Material** & **Wave Speed C\({}_{\text{P}}\) (m/s)** & **Density, \(\rho\) (kg/m\({}^{3}\))** & **Impedance, Z (MRayl)** \\ \hline Buffer material & 2508 & 1652 & 4.14 \\ Formwork & 2000 & 1800 & 3.60 \\ Concrete 1 (fresh) & 1095 & 2500 & 2.74 \\ Concrete 2 & 1341 & 2500 & 3.35 \\ Concrete 3 (similar impedance to buffer) & 1673 & 2500 & 4.18 \\ Concrete 4 & 1788 & 2500 & 4.47 \\ Concrete 5 & 2097 & 2500 & 5.24 \\ Concrete 6 & 2300 & 2500 & 5.75 \\ Concrete 7 & 2683 & 2500 & 6.71 \\ Concrete 8 & 3098 & 2500 & 7.75 \\ Concrete 9 & 3633 & 2500 & 9.08 \\ Concrete 10 (hardened) & 4049 & 2500 & 10.12 \\ \hline \end{tabular}
\end{table} 
Table 1: Properties of materials used in numerical simulation. -fresh concrete) a reflection is visible because of the impedance mismatch at the interface, as aforementioned. For case #3 (black line) however, this cycle is reasonably suppressed because of similar impedances for concrete and buffer (both approximately 4.15 MRayl). For the last case of hardened concrete (#10, blue line) a reflection is again obvious and even stronger than case #1 while now the phase of the cycle is reversed. This is because the concrete has become stiffer than the buffer and therefore, the reflection coefficient mathematically obtains the opposite sign [17]. By using the high frequency of 1 MHz, the reflections can be detected showing in the fresh state a weak reflection, while for the next stage (#3) the reflection can hardly be identified. Later, for concrete stage #10, the reflection amplitude climbs up to a value 5 to 6 times higher than the minimum of case #3, allowing clear observation, as seen in Figure 11b that focuses on the reflection arrival.