# Hybrid Shielding for Hypervelocity Impact of Orbital Debris on Unmanned Spacecraft
## 2 Numerical Modelling
### Finite Elements and SPH Model  
The finite-element method is used to model the impact response of the whole system, including the front (bumper) and back plates, and all models were set up in LS-DYNA.  
Solid constant stress solid elements are used to model the bumper plate and black wall, and four-node constant thickness shell elements are used in the honeycomb structure. The same mesh size is used in both the bumper and back plates and a thorough mesh convergence analysis is carried out to determine the optimal mesh size. The spherical projectile and the impact area on the front plate are modelled using Smoothed Particle Hydrodynamics (SPH). The convergence analysis was also extended into the SPH domain to determine the optimal particle density, especially as this method is highly computationally heavy and often leads to high CPU times. Specific contacts were implemented to model the interaction between the different model components: (i) tied contacts between the finite element and the SPH particles in the bumper plate, and (ii) automatic contacts between all SPH particles--both from the impactor and the impacted area--and all other finite-element components in the model (the back plate, the front plate and the honeycomb). In LS-DYNA, these contacts were implemented with the keywords *CONTACT_AUTOMATIC_NODES_TO_SURFACE and *CONTACT_TTED_NODES_TO_SURFACE, respectively.  
To optimise computational efficiency the proposed models explored symmetries when possible, that is, when all impact, geometrical and boundary conditions were symmetrical. \(Oxz\) and \(Oyz\) are the two symmetry planes of this model, as can be seen in Figure 5a. One of these planes--symmetry plane \(Oxz\)--is used in this research, as shown in Figure 5b. Symmetry plane \(Oyz\), however, cannot be used, as it is not a symmetry plane for the impact location analyses, where symmetry is broken when the debris particle impacts at different locations.  
The HVI problem being analysed is a kinematics-dominated problem; thus, imposing fixed boundary conditions on the Whipple plates is not strictly necessary. This is supported by the progression of the post-impact radial stress wave on the Whipple front plate. To  
Figure 5: Simulation set-up showing (**a**) FE mesh, SPH regions and coordinate system; and (**b**) boundary conditions.  
ensure the consistency of analysis with the experimental results of [PERSON] et al. [9], these boundary conditions were imposed. The implemented boundary conditions are shown in Figure 5. The external edges of the finite-element meshes on the front and back plate, as well as the honeycomb core, are fixed in all degrees of freedom (linear and rotational). The motion of all SPH particles on the plane of symmetry was also restricted to in-plane motion, that is, fixed in the y-direction, as shown in Figure 5.  
An initial constant velocity of 6500 m/s is imposed on the spherical projectile along the direction perpendicular to the bumper plate (\(z\)-direction in Figure 5). This velocity is representative of a generic micro-meteoroid orbital debris particle impacting the shield.