# Numerical Identification of Material Model Parameters of UHPFRC Slab under Blast Loading
## 6 Inverse Identification
337 (aggregate size, fracture energy, Poisson's number, modulus of elasticity, compressive strength, tensile strength) therefore represent a good candidate for use in simulations of fast dynamic events using the Winfrith concrete material model in the LS-Dyna software environment. Figure 8: Optimization chain of the material model parameter in the software environment Optislang. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter’s** & **Aggregate** & **Fracture** & **Poisson** & **Tangent** & **Compressive** & **Tensile** \\
**Name** & **Size** & **Energy** & **Ratio** & **Modulus** & **Strength** & **Strength** \\
**(Unit)** & **(mm)** & **(kN/mm)** & **(-)** & **(GPa)** & **(GPa)** & **(GPa)** \\ \hline Range & 0.25–10 & 0.001–0.02 & 0.12–0.3 & 15–50 & 0.03–0.12 & 0.004–0.012 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Range of values of concrete parameters used in the optimizations. Results of the sensitivity analysis representing the degree of response of the output parameter to the input parameters can be quantified using the [PERSON]'s correlation coefficient [68,69]. This coefficient for population represents the standardized covariance between two random variables, while when applied to a sample is defined as:  
\[r_{xy}\approx\ \frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{ \sqrt{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\sqrt{\sum_{i=1}^{n}(y_{i}- \overline{y})^{2}}} \tag{10}\]  
where \(r_{xy}\) is Pearson's correlation coefficient for the sample, \(n\) is sample size, \(x_{i},y_{i}\) are the individual sample points indexed with \(i\) and \(\overline{x}=\frac{1}{n}\sum_{i=1}^{n}(x_{i})\) is the sample mean (analogously for \(y\)). The resulting values of Pearson's correlation coefficient \(r_{xy}\) shown in Table 11 show a high, medium and low degree of dependence of the output parameter-objective function O on three input parameters' fracture energy, tensile strength and tangent modulus, respectively [68,69]. These values of correlation coefficients \(r_{xy}\) indicate [67,70] that these parameters are closely related to the physical essence of the investigated problem, i.e., bending of the body as a result of loading shock wave. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Vector** & **Aggregate Size** & **Fracture** & **Poisson** & **Tangent** & **Compressive Strength** & **Tensile Strength** & **O** \\
**Number** & **(mm)** & **Energy** & **Ratio** & **(-)** & **Modulus** & **(GPa)** & **(GPa)** & **(GPa)** & **(mm)** \\ \hline
337 & 9.6283 & 0.0016 & 0.1590 & 44.8925 & 0.0659 & 0.0070 & 0.840 \\
141 & 9.6770 & 0.0014 & 0.1347 & 42.1475 & 0.0709 & 0.0083 & 1.023 \\
246 & 7.8367 & 0.0022 & 0.2128 & 47.3675 & 0.0500 & 0.0060 & 1.144 \\ \hline \end{tabular}
\end{table} 
Table 10: Resulting design values of concrete parameters. Figure 9: Comparison of the deflection curves obtained by simulation (red and grey) with the experimentally obtained curve (green colour). Where \(r_{xy}<size,O>\) is Pearson's correlation coefficient for aggregate size and Objective function \(O\) (analogously for fracture energy \(G_{f}\), Poisson's number \(\eta\), tangent modulus \(E\), compressive strength \(f_{c}\) and tensile strength \(f_{l}\)). On the other hand, parameters, such as aggregate size, Poisson's ratio and compressive strength, affect the deflection to a lesser extent and their values can be chosen as constants in the future when solving optimization problems of a similar nature due to the reduction of the computing time demands.