# Determination of Concrete Formwork Removal Time Based on Ultrasound Reflection
## 5 Numerical Simulations  
In order to increase the understanding of the phenomenon and validate the explanation of the results, numerical simulations were also conducted. A certain issue that deserved specific attention is that in the experiment, between the buffer material and concrete a third material is present (glass polyester composite of the formwork). Even though it is much thinner (3 mm) than the buffer (50 mm) and concrete column (diameter 200 mm), and although its properties do not differ much from the buffer, it might still exercise a strong influence, and this cannot be simply assessed by Equation (1). Therefore, the exact cross-section of the specimen was modeled with the material of the formwork. Numerical simulations were conducted with a commercially available software Wave2000 [23]. It computes displacement vectors by solving 2D elastic wave equations using a method of finite differences. The specific acoustic equation that is simulated is:  
\[\rho\frac{\theta^{2}u}{\theta t^{2}}=\bigg{(}\mu+\eta\frac{\theta}{\theta t} \bigg{)}\
abla^{2}u+\bigg{(}\lambda+\mu+\varphi\frac{\theta}{\theta t}+\frac{ n}{3}\frac{\theta}{\theta t}\bigg{)}\
abla(\
abla u), \tag{4}\]  
where \(u\) is the displacement vector (consisting of two \(u_{x}\) and \(u_{y}\) components perpendicular to each other), \(\rho\) is the density (kg/m\({}^{3}\)), \(\lambda\) and \(\mu\) are the first and second Lame constants (Pa), \(\eta\) and \(\varphi\) are the \"shear\" and \"bulk\" viscosity (Pa\(\cdot\)s) and \(t\) is time (s) [23]. The model contains three media with distinct sets of material properties. The buffer material has a minimum width of 50 mm (at the center). It is placed adjacent to the formwork from one side, and from the other side, a source and a receiver transducer are placed with a distance of 50 mm between them (see Figure 8). The concrete is placed in a semi-circle with a diameter of 200 mm. An infinite boundary condition is used on the right side since only the first reflection at the interface between the buffer material and the concrete is of interest. The resolution in time was 0.05 \(\upmu\)s, sufficiently shorter than the period of the wave even for the highest frequency tested (1 MHz). Therefore, each cycle was represented by as many as 20 points, according to relevant recommendations in literature [24]. Similarly, the space resolution was set to 0.15 mm, which is much smaller than the wavelength considering the wave speed of the buffer material (approximately 3 mm for 1 MHz). To simulate the development of stiffness in the fresh concrete, several stages of the concrete evolution were defined to account for the ongoing hydration process (see Table 1). The properties of the buffer material are taken similar to the experimental. The formwork is made of fiber glass with a polyester matrix and has a density of 1800 kg/m\({}^{3}\). The ultrasonic  
Figure 8: Numerical displacement field (**a**) 20 \(\upmu\)s, (**b**) 30 \(\upmu\)s and (**c**) 40 \(\upmu\)s after excitation of 1 MHz for concrete properties #1 (fresh concrete). speed C\({}_{\text{P}}\) was measured using the ultrasonic velocity meter to the value of 2000 m/s through the thickness of the material. The material properties are illustrated in Table 1.  
It is noted that the impedance of the formwork is not far apart from the one of the buffer. The source or pulser has a size of 15 mm which is equivalent to the diameter of the transducers used in the experiments. The displacement source wave function is defined as a sinusoidal cycle that is excited vertical to the surface, and therefore with horizontal direction in Figure 8a. Since the simulations are in the linear regime, the unit of the amplitude does not play a role. If one assumes that the excitation is in nm, this is the unit of the amplitude of the receiver that is presented. While the experimental frequency was 150 kHz, the frequency of 1 MHz, was also simulated.