# Analysis of a Collision-Energy-Based Method for the Prediction of Ice Loading on Ships
## 2 Background
### Contact Geometry  
To calculate ice loads using the Popov Method, the ship's indentation into the ice must be defined. Following Tunik [11], the indentation can be assessed based on the assumed form of the contact area between the ship and the ice, which depends both on the form of the ship's hull, and on the form of the ice at the point of impact. Per the Popov Method, this form is modelled in terms of an ice-hull overlap geometry, in the following referred to as contact geometry [34].  
In real-life ship-ice interactions, many different types of contact geometries occur. Thus, to enable the assessment of ice loading from different types of ship-ice interactions, [PERSON] et al. [9], [PERSON] [34] and [PERSON] [35] present contact geometry models corresponding to different ship-ice interactions. A selection of these contact geometries are presented in Figure 1, and explained in the following:  
* Round contact geometry. This model, which is behind the design scenario of the ice class standards of the Russian Maritime Register of Shipping (RMRS), corresponds to an oblique collision with a rounded ice edge [12]. Related geometrical parameters are the radius of ice floe \(\mathrm{R},[m]\), the waterline angle \(\alpha,[deg]\) and normal frame angle \(\beta^{\prime},[deg]\). Parameters \(\alpha\) and \(\beta^{\prime}\) are dependent on the hull form, whereas \(\mathrm{R}\) is dependent on the ice conditions. In the RMRS ice class rules, \(\mathrm{R}\) is assumed at 25 \(m\).
* Angular (wedge) contact geometry. This model is applied in the design scenario of the PC rules [14]. Related geometrical parameters are the ice edge opening angle \(\phi,[deg]\), waterline angle \(\alpha,[deg]\) and normal frame angle \(\beta^{\prime},[deg]\). According to [PERSON], et al. [9], \(\phi\) can be determined by observation of ice segments broken by a ship. Alternatively, \(\phi\) can be determined by numerical simulations [36].
* Symmetric v wedge contact geometry. This model can be applied to assess ice loads from beaching impacts, i.e., when a ship is ramming an ice sheet so that the bow of the ship rises upwards onto the ice sheet [34]. Related geometrical parameters are waterline angle \(\alpha,[deg]\), frame angle \(\beta,[deg]\) and stem angle \(\gamma\), \([deg]\), all of which depend on the hull form at the point of impact.
* Right-apex oblique contact geometry. This model, defined by [PERSON] [34], corresponds to continuous operation in ice without ramming. Related geometrical parameters are waterline angle \(\alpha,[deg]\), normal frame angle \(\beta^{\prime},[deg]\), stem angle \(\gamma,[deg]\) and frame angle \(\beta,[deg]\), all of which depend on the hull form at the point of impact.  
* Spherical contact geometry. This model can be applied for the assessment of ice loads on a bulbous bow [20]. Related geometrical parameters are the contact radius \(\mathbf{r},[m]\), waterline angle \(\alpha,[deg]\) and the normal frame angle \(\beta^{\prime},[deg]\), all of which depend on the hull form at the point of impact.
* Pyramid contact geometry. This model can be applied to assess ice loads from ship-iceberg interactions resulting in structural deformations [35]. Related geometrical parameters are the ice edge opening angle \(\phi,[deg]\), waterline angle \(\alpha,[deg]\) and normal frame angle \(\beta^{\prime},[deg]\).  
As per the contact geometries presented above, the level of ice loading might depend on the hull form in terms of the frame angle \(\beta\), waterline angle \(\alpha\) and stem angle \(\gamma\), as shown in Figure 1. Generally, a reduced frame angle, an increased waterline angle, or an increased stem angle result in higher ice loads, and vice versa. However, as discussed by [PERSON], et al. [9] the influence of the different angles depends upon the point of impact.