## 3 Meshless Simulations
### Numerical Modelling  
Drop impact tests on EVA square pads having thickness equal to 50 and 70 mm were simulated numerically using the commercial code LS-DYNA. The calculation grids of the pads and the missile, as well as the background element mesh needed to carry on the Gaussian numerical quadrature, were built using the pre-processor HYPERMESH, starting from 3D models generated in the Solidworks geometrical modelling environment. In order to reduce the computational burden, the drop assembly geometry was simplified by considering only the hemispherical missile as an impacting object (with a mass equal to the fall group mass). The hemispherical missile, made of aluminium alloy, was discretized using tetrahedral elements with an average size equal to 10 mm. Its deformations during the interaction with the pads were negligible compared to those of the pad. For this reason, the missile was modelled as a rigid body using the function MAT_RIGID (*MAT_020) [22]. To generate the nodes grid and the background mesh, the pads were divided into two distinct volumes and different node spacing was adopted for each volume. The first volume is the \"impact zone,\" which has a prismatic shape with a base of size 200 mm \(\times\) 200 mm, a height equal to the pad thickness, and includes the foam material more severely deformed by the missile actions (Figure 3). This volume was meshed with small hexahedral elements having edge size equal to 5 mm. The remaining portion of the pad was instead discretized with tetrahedral elements having a gradually increasing size from 5 mm at the inner boundary to around 10 mm at the external edges of the pad. The MAT_LOW_DENSITY_FOAM (*MAT57) [22] was used as the foam constitutive model. This model is defined in terms of a foam compression curve, the Young's modulus, the Poisson ratio, and some constitutive parameters defining the hysteretic and viscoelastic aspects of the material response. The foam unloading behaviour was controlled by the hysteretic unloading factor, HU, and the SHAPE parameter, both reported in Table 1. HU ranged in value between zero and one. Low values of HU shifted the unloading path downward. The lowest value of HU did not sufficiently account for energy loss, so the SHAPE factor was adjusted to increase energy loss by further shifting the unloading curve down. Viscous effects were controlled by a DAMP factor. Increasing the DAMP values is equivalent to adding a material damper. The suggested range of DAMP was between 0.05 and 0.5. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Material Models**} & **Density, \(\rho\)** & \multirow{2}{*}{**E (GPa)**} & \multirow{2}{*}{**v**} & **HU** & **DAMP** & **SHAPE** \\  & **(kg/mm3)** & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material models used in the simulation and constitutive parameters. HU: hysteretic unloading factor; DAMP: viscous coefficient; SHAPE: shape factor for unloading. Figure 3: EFG model: Pad foam with the impact zone highlighted in green (**Left**) and missile (**Right**). The values of previous foam parameters were obtained by 'tuning' the results of a series of numerical simulations of the compressive tests with those obtained by the real compressive tests. A comparison among experimental results and numerical predictions carried out with optimal values of these constitutive parameters is given in Figure 2. Experimental data and numerical outcomes are in good agreement for the whole loading curves and for most of the unloading ones. The numerical model did not properly match the final part of the experimental unloading curves, since a part of the foam rearrangement takes place on a time scale greater than the compressive test time scale. However, in the following it will be shown that this does not influence the model accuracy in reproducing missile impacts on foam pads. During the impact testing, penetration depths greater than the 80% of the pad thickness were observed.