# Numerical Identification of Material Model Parameters of UHPFRC Slab under Blast Loading
## 6 Inverse Identification  
A one-step inverse identification method was used to find the values of concrete material parameters for use in simulations of fast dynamic events. The design vector of the Winfrith concrete material model in the LS-Dyna software environment, which is composed of six input parameters, can be written in the form:  
\[X=\left\{E,\eta,f_{c},f_{t},G_{f},size\right\}, \tag{7}\]  
where \(X\) is the design vector, \(E\) (GPa) is the tangent modulus, \(\eta\) (-) is Poisson's number, \(f_{c}\) (GPa) is the compressive strength, \(f_{t}\) (GPa) is the tensile strength, \(G_{f}\) (kN/mm) is the fracture energy and \(size\) (mm) is the aggregate size. These parameters enter the simulation as variables, the other parameters of the material model were entered as constants, i.e., the density of the concrete and Young's modulus and yield strength of the fibre reinforcement (Chapter 5). Based on this, the values of the deflections obtained by simulation (Figure 7) at time instants can be written as a function of the design vector:  
\[y_{i}^{s}=f\{X\}, \tag{8}\]  
where \(y_{i}^{e}\) (mm) is the value of plate deflection over time. The goal of inverse parameter identification can subsequently be defined as minimizing the value of the objective function. The objective function is defined as the square root of the mean of the squares of the deviations of the deflection values obtained experimentally and by means of simulation, i.e.,:  
\[O=\sqrt{\frac{\sum_{i=1}^{n}\left(y_{i}^{s}-y_{i}^{e}\right)^{2}}{n}}, \tag{9}\]  
where \(O\) (mm) is the objective function and \(y_{i}^{s}\) (mm) and \(y_{i}^{e}\) (mm) are the deflection values obtained by simulation, respectively, by the experiment at the same time points and n is the number of values. An optimization chain for sensitivity analysis was created for this purpose in the Optislang program environment (Figure 8, more in depth theory and design of optimisation process can be found in [66]). In this chain, a set of vectors of input parameters of the material model were generated using the Advanced Latin Hypercube stochastic (ALH) method [67]. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Parameter’s** & **Aggregate** & **Fracture** & **Poisson** & **Tangent** & **Compressive** & **Tensile** \\
**Name** & **Size** & **Energy** & **Ratio** & **Modulus** & **Strength** & **Strength** \\
**(Unit)** & **(mm)** & **(kN/mm)** & **(-)** & **(GPa)** & **(GPa)** & **(GPa)** \\ \hline Parameter’s value & 5.0 & 0.001 & 0.2 & 35 & 0.06 & 0.006 \\ \hline \end{tabular}
\end{table} 
Table 8: Initial seed vector of input parameters’ values of the Winfrith material model used in the simulation. The first component named \"Slab.k\" is used to enter the input parameters that were to be optimized. From the text input file of the LS-Dyna solver, the position and values of six of these input parameters of the concrete material model were found: modulus of elasticity, Poisson's number, tensile and compressive strength, fracture energy and aggregate fraction size. Their ranges were set in the main menu, as shown in Table 9, based on which 800 design vectors were generated using the ALH stochastic method, covering the entire design space of input parameters [67]. In the \"LS-Dyna Solver\" component itself, a simulation was run sequentially for these 800 different design parameter vectors. Using the \"Simulation Output\" component, the resulting deflection curve was obtained. Subsequently, in the \"Comparison\" component, the value of the objective function \(O\) was calculated according to Equation (9) from two input files, the reference curve, obtained by the experiment and from the output file from simulation for each design vector (Figure 9). Table 10 shows the parameter values of the three design vectors with the lowest objective function obtained by sensitivity analysis from 800 design vectors, as well as the value of the objective function O. The parameter values of design vector No.